Laplace_Transforms.ipynb

{
 "metadata": {
  "name": "Laplace_Transforms"
 },
 "name": "Laplace_Transforms",
 "nbformat": 3,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "%load_ext sympyprt",
     "language": "python",
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "from sympy import *",
     "language": "python",
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "source": "Examples of Fourier transformations."
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "p = Symbol(\"p\")\nt = Symbol(\"t\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "n = Symbol(\"n\", integer=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "a = Symbol(\"a\")\nb = Symbol(\"b\")\nc = Symbol(\"c\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "source": "The following examples are taken from 'Bronstein', 6th edition, page 1097ff"
    },
    {
     "cell_type": "markdown",
     "source": "1"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "0",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 6,
       "text": "0"
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 7,
       "text": "(0, -\u221e, True)"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAMBAMAAABCcoqQAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAVJlmiUTNuxAydkvU/DYAAABBSURBVAgd\nY2AQMhFgYGBNYChlYGAzYDBjYOgWYIhuYJgdwDB7AkP0BBADQkIYQFnpBgb2BSCVQF2pDAwMIlYC\nDADyLg3mxPBcdQAAAABJRU5ErkJggg==\n",
       "prompt_number": 8,
       "text": "0"
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "source": "2"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 9,
       "text": "1"
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 10,
       "text": "\n\u239b1         \u239e\n\u239c\u2500, 0, True\u239f\n\u239dp         \u23a0"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGIAAAASBAMAAABSqRb8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\niL8mYZPpBTOk0gAAAABJRU5ErkJggg==\n",
       "prompt_number": 11,
       "text": "Heaviside(t)"
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "source": "3"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "t**(n-1)/factorial(n-1)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAsBAMAAADIs8jyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nsMgPn890LGewX4kAAAAASUVORK5CYII=\n",
       "prompt_number": 12,
       "text": "\n  n - 1 \n t      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n(n - 1)!"
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 13,
       "text": "\n\u239b -n               \u239e\n\u239dp  , 0, -n + 1 < 1\u23a0"
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIkAAAAsBAMAAACj24anAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7UMNHPSGci787m9YfgChjidYFjdW5QAAAABJRU5ErkJggg==\n",
       "prompt_number": 14,
       "text": "\n n - 1             \nt     \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        \u0393(n)       "
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "source": "4"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "t**(n-1)/factorial(n-1)*exp(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAsBAMAAADIs8jyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 15,
       "text": "\n n - 1  a\u22c5t\nt     \u22c5\u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  (n - 1)! "
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#laplace_transform(_, t, p)\n# unsupported operand type(s) for *: 'NoneType' and 'Mul'",
     "language": "python",
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/(p-a)**n",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAASBAMAAAAteAQNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqsiVN12RBDvzTKJ\nu5m05BpfAAABRElEQVQoFWNgYBBgwANgkhxJhjlAZUwGeNQy9EElWVgT+IFMEXxqGVgSGBjfvXv3\nkKGfYStQYQCm4i1IQiugbF+GMwwgrRjAEUkkEcp+xrA6gYGpAEkGykRWvAsqpsxwrIAB5GwoYEt0\nhbAgilnyGuUakOUZgA6Hg2sMT5EVN/Yn8D1g4GyAyzMwWMLZjAEM75AVV+xn4AlgYEUOWVMGBua7\nIJDAeYAhBKiY7+7dpXfvKgBZuQy8FxhYN0AMAJOdcLZ9AccPCAfqwYsMQP0ozkB4cH8DY4ADWDVU\n8WeGSjQPcsGt4TSwUkhAUsz9G+SsvWARKMEGkQfy2B9KqEF8AzGZ8aIM0CBBZMUcAcg8CLsETDEl\ngChYdIOFGI5DKEySswAoxn4ARYKpAYWL4NiDJJoQfBCLIwGVD+Oxr38BZGbAuCAaAI5KQJsjLaVr\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 17,
       "text": "\n        -n\n(-a + p)  "
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKIAAAAYBAMAAACLjTs7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
       "prompt_number": 18,
       "text": "\n                       \u239b        -n            \u239e\nInverseLaplaceTransform\u239d(-a + p)  , p, t, None\u23a0"
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "source": "5"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(exp(b*t)-exp(a*t))/(b-a)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAqBAMAAAAAIxtcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSNtfPMxfLIjibZZ14m9TOjQAvgGlQnqF/luUsQAAAABJRU5ErkJggg==\n",
       "prompt_number": 19,
       "text": "\n   a\u22c5t    b\u22c5t\n- \u212f    + \u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    -a + b   "
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 20,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/((p-a)*(p-b))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIwAAAApBAMAAAAVP9xQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nLyOTCQ7VaemH7uNjdwI8jEP+APiPy8gm+NIrAAAAAElFTkSuQmCC\n",
       "prompt_number": 21,
       "text": "\n        1        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n(-a + p)\u22c5(-b + p)"
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAm0AAAAqBAMAAAAT53FwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJ18Y7H8BARUNlR5SnCoAAAAASUVORK5CYII=\n",
       "prompt_number": 22,
       "text": "\n\u239b   t\u22c5re(a)                 t\u22c5re(a)                   t\u22c5re(b)                 \n\u239d\u2148\u22c5\u212f       \u22c5sin(t\u22c5im(a)) + \u212f       \u22c5cos(t\u22c5im(a)) - \u2148\u22c5\u212f       \u22c5sin(t\u22c5im(b)) - \u212f\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                     a - b                    \n\nt\u22c5re(b)             \u239e             \n       \u22c5cos(t\u22c5im(b))\u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                  "
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "source": "6"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(b*exp(b*t)-a*exp(a*t))/(b-a)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAqBAMAAABRtyg/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nLnztlr7JOKOb84SLuD4H+v4DYKqYYrhT1mMAAAAASUVORK5CYII=\n",
       "prompt_number": 23,
       "text": "\n     a\u22c5t      b\u22c5t\n- a\u22c5\u212f    + b\u22c5\u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n      -a + b     "
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 24,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "p/((p-a)*(p-b))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIwAAAAlBAMAAABi/RwrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
       "prompt_number": 25,
       "text": "\n        p        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n(-a + p)\u22c5(-b + p)"
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAApAAAAAqBAMAAAD2VXZZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 26,
       "text": "\n\u239b     t\u22c5re(a)                   t\u22c5re(a)                     t\u22c5re(b)           \n\u239d\u2148\u22c5a\u22c5\u212f       \u22c5sin(t\u22c5im(a)) + a\u22c5\u212f       \u22c5cos(t\u22c5im(a)) - \u2148\u22c5b\u22c5\u212f       \u22c5sin(t\u22c5im(b\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                         a - b                \n\n        t\u22c5re(b)             \u239e             \n)) - b\u22c5\u212f       \u22c5cos(t\u22c5im(b))\u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                          "
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "source": "7"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-a*t)/sqrt(b**2-a**2)*sin(sqrt(b**2-a**2)*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAK8AAAA6BAMAAAA0FxS2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nI58WGk5tlOINvFa5yNTpRvh/RfZ/q/YA0XCYkinjRIYAAAAASUVORK5CYII=\n",
       "prompt_number": 27,
       "text": "\n         \u239b     ___________\u239e\n -a\u22c5t    \u239c    \u2571    2    2 \u239f\n\u212f    \u22c5sin\u239dt\u22c5\u2572\u2571  - a  + b  \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          ___________      \n         \u2571    2    2       \n       \u2572\u2571  - a  + b        "
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPAAAAA6BAMAAAB8CMlzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAATzIqz/AFtq/6ty5iZsAAAAAElFTkSuQmCC\n",
       "prompt_number": 28,
       "text": "\n                \u239b         \u239b     ___________\u239e      \u239e\n                \u239c -a\u22c5t    \u239c    \u2571    2    2 \u239f      \u239f\nLaplaceTransform\u239d\u212f    \u22c5sin\u239dt\u22c5\u2572\u2571  - a  + b  \u23a0, t, p\u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                      ___________                  \n                     \u2571    2    2                   \n                   \u2572\u2571  - a  + b                    "
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/(p**2+2*a*p+b**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAApBAMAAAAvw6FmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 29,
       "text": "\n       1       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         2    2\n2\u22c5a\u22c5p + b  + p "
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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       "prompt_number": 30,
       "text": "\n\u239b           __________________________________________________________________\n\u239c          \u2571                                                                  \n\u239c       4 \u2571    4          2      \u239b 2\u239e       2      2                      \u239b 2\u239e\n\u239c     t\u22c5\u2572\u2571   re (a) - 2\u22c5re (a)\u22c5re\u239db \u23a0 + 2\u22c5re (a)\u22c5im (a) - 4\u22c5re(a)\u22c5im(a)\u22c5im\u239db \u23a0\n\u239c                                                                             \n\u239c- \u2148\u22c5\u212f                                                                        \n\u239d                                                                             \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n________________________________________________                              \n     2                                    2         \u239b     \u239b                  \u239b\n      \u239b 2\u239e       \u239b 2\u239e   2        4         \u239b 2\u239e     \u239catan2\u239d2\u22c5re(a)\u22c5im(a) - im\u239d\n + re \u239db \u23a0 + 2\u22c5re\u239db \u23a0\u22c5im (a) + im (a) + im \u239db \u23a0 \u22c5cos\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                    \u239d                         \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                                                                           ___\n 2\u239e    2        \u239b 2\u239e     2   \u239e\u239e                                           \u2571   \nb \u23a0, re (a) - re\u239db \u23a0 - im (a)\u23a0\u239f                                        4 \u2571    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f    \u239b              \u239b   _________\u239e\u239e    t\u22c5\u2572\u2571   re\n  2                           \u23a0    \u239c              \u239c  \u2571  2    2 \u239f\u239f             \n                               \u22c5sin\u239dt\u22c5im(a) - t\u22c5im\u239d\u2572\u2571  a  - b  \u23a0\u23a0 + \u212f         \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n______________________________________________________________________________\n                                                                    2         \n4          2      \u239b 2\u239e       2      2                      \u239b 2\u239e      \u239b 2\u239e     \n (a) - 2\u22c5re (a)\u22c5re\u239db \u23a0 + 2\u22c5re (a)\u22c5im (a) - 4\u22c5re(a)\u22c5im(a)\u22c5im\u239db \u23a0 + re \u239db \u23a0 + 2\u22c5\n                                                                              \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n_________________________________                                             \n                           2         \u239b     \u239b                  \u239b 2\u239e    2       \n  \u239b 2\u239e   2        4         \u239b 2\u239e     \u239catan2\u239d2\u22c5re(a)\u22c5im(a) - im\u239db \u23a0, re (a) - r\nre\u239db \u23a0\u22c5im (a) + im (a) + im \u239db \u23a0 \u22c5cos\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                     \u239d                           2            \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                                                                              \n \u239b 2\u239e     2   \u239e\u239e                                                              \ne\u239db \u23a0 - im (a)\u23a0\u239f                                             \u239b   _________\u239e   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f    \u239b              \u239b   _________\u239e\u239e           \u239c  \u2571  2    2 \u239f   \n               \u23a0    \u239c              \u239c  \u2571  2    2 \u239f\u239f      -t\u22c5re\u239d\u2572\u2571  a  - b  \u23a0   \n                \u22c5cos\u239dt\u22c5im(a) - t\u22c5im\u239d\u2572\u2571  a  - b  \u23a0\u23a0 + \u2148\u22c5\u212f                   \u22c5si\n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n \u239b      ______________________________________________________________________\n \u239c     \u2571                                                                      \n \u239c  4 \u2571    4          2      \u239b 2\u239e       2      2                      \u239b 2\u239e    \nn\u239ct\u22c5\u2572\u2571   re (a) - 2\u22c5re (a)\u22c5re\u239db \u23a0 + 2\u22c5re (a)\u22c5im (a) - 4\u22c5re(a)\u22c5im(a)\u22c5im\u239db \u23a0 + r\n \u239d                                                                            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                             2\n\n                                                                              \n                                                                              \n____________________________________________                                  \n 2                                    2         \u239b     \u239b                  \u239b 2\u239e \n  \u239b 2\u239e       \u239b 2\u239e   2        4         \u239b 2\u239e     \u239catan2\u239d2\u22c5re(a)\u22c5im(a) - im\u239db \u23a0,\ne \u239db \u23a0 + 2\u22c5re\u239db \u23a0\u22c5im (a) + im (a) + im \u239db \u23a0 \u22c5sin\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                \u239d                           2 \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   _________________                                                          \n\u22c5\u2572\u2571 (a - b)\u22c5(a + b)                                                           \n\n                                                                              \n                                                                              \n                                     \u239e         \u239b   _________\u239e    \u239b      ______\n   2        \u239b 2\u239e     2   \u239e\u239e          \u239f         \u239c  \u2571  2    2 \u239f    \u239c     \u2571      \n re (a) - re\u239db \u23a0 - im (a)\u23a0\u239f          \u239f    -t\u22c5re\u239d\u2572\u2571  a  - b  \u23a0    \u239c  4 \u2571    4  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + t\u22c5im(a)\u239f - \u212f                   \u22c5cos\u239ct\u22c5\u2572\u2571   re (a\n                          \u23a0          \u23a0                           \u239d            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n______________________________________________________________________________\n                                                                 2            \n        2      \u239b 2\u239e       2      2                      \u239b 2\u239e      \u239b 2\u239e       \u239b\n) - 2\u22c5re (a)\u22c5re\u239db \u23a0 + 2\u22c5re (a)\u22c5im (a) - 4\u22c5re(a)\u22c5im(a)\u22c5im\u239db \u23a0 + re \u239db \u23a0 + 2\u22c5re\u239d\n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n______________________________                                                \n                        2         \u239b     \u239b                  \u239b 2\u239e    2        \u239b \n 2\u239e   2        4         \u239b 2\u239e     \u239catan2\u239d2\u22c5re(a)\u22c5im(a) - im\u239db \u23a0, re (a) - re\u239db\nb \u23a0\u22c5im (a) + im (a) + im \u239db \u23a0 \u22c5sin\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                  \u239d                           2               \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                        \u239e          ___________________________________________\n                        \u239f         \u2571                                           \n                       \u239e\u239f      4 \u2571    4          2      \u239b 2\u239e       2      2   \n2\u239e     2   \u239e\u239e          \u239f\u239f  - t\u22c5\u2572\u2571   re (a) - 2\u22c5re (a)\u22c5re\u239db \u23a0 + 2\u22c5re (a)\u22c5im (a)\n \u23a0 - im (a)\u23a0\u239f          \u239f\u239f                                                     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + t\u22c5im(a)\u239f\u239f\u22c5\u212f                                                   \n            \u23a0          \u23a0\u23a0                                                     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n_______________________________________________________________________       \n                            2                                    2         \u239b  \n                   \u239b 2\u239e      \u239b 2\u239e       \u239b 2\u239e   2        4         \u239b 2\u239e     \u239cat\n - 4\u22c5re(a)\u22c5im(a)\u22c5im\u239db \u23a0 + re \u239db \u23a0 + 2\u22c5re\u239db \u23a0\u22c5im (a) + im (a) + im \u239db \u23a0 \u22c5cos\u239c\u2500\u2500\n                                                                           \u239d  \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                                                                              \n   \u239b                  \u239b 2\u239e    2        \u239b 2\u239e     2   \u239e\u239e                 \u239b   ___\nan2\u239d2\u22c5re(a)\u22c5im(a) - im\u239db \u23a0, re (a) - re\u239db \u23a0 - im (a)\u23a0\u239f                 \u239c  \u2571  2\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f - t\u22c5re(a) + t\u22c5re\u239d\u2572\u2571  a \n                         2                           \u23a0                        \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                    \n______\u239e             \n    2 \u239f             \n - b  \u23a0             \n                    \n       \u22c5Heaviside(t)\n                    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                    \n                    "
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "source": "8"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sin(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAASBAMAAAANlFvwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZlTNRImrEN27MiJ2\n75lZwTvPAAABJElEQVQoFV2QvUrDUBTHf0ljorElVsGCKBRdxE0QdAyI4CQdfAAfQEgmN23wCRxF\nHC4IglODLi6lyeRzuDkUURSEKuhJmpu2Hrjn/D8u5557YDLmSjpCWrJjjTgr0EqopfkCzCrMZIjX\nSjMqzJkWbBdYF30bz4dUq0W1RcujI/kGjNUF91Ltftf7wj051XSdgx1hQvaww7ZyfujJw20xr3nJ\nEtRC9n0LT/GeJbrSKeIVBmJOx9i/V5k+yGd4lPstznHfMlMR3gdxaZ5Cz3e+MC6GbZeYSkpTZuiE\nRtS0EzMfaJGqL+Zn3raiqMUPR0ml2YA7WE5TN9i6/dgINuUbCdZz4zg2+wrq0no8nGiM/V8fhyPT\nkvVOhi2bKOJEg7I6SQmf4A9IJ0QQle9nVwAAAABJRU5ErkJggg==\n",
       "prompt_number": 31,
       "text": "sin(a\u22c5t)"
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 32,
       "text": "\n\u239b   a        \u2502                 \u239b          2      \u239e\u2502    \u239e\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 = 0\u239f\n\u239c 2    2                                               \u239f\n\u239da  + p                                                \u23a0"
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAApkAAAAlBAMAAAD7Fu9GAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 33,
       "text": "\n   \u239b   2\u22c5t\u22c5im(a)                 2\u22c5t\u22c5im(a)                                    \n-\u2148\u22c5\u239d\u2148\u22c5\u212f         \u22c5sin(t\u22c5re(a)) - \u212f         \u22c5cos(t\u22c5re(a)) + \u2148\u22c5sin(t\u22c5re(a)) + cos\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                       2                      \n\n         \u239e  -t\u22c5im(a)             \n(t\u22c5re(a))\u23a0\u22c5\u212f        \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                 "
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "source": "9"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sin(a*t+b)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFUAAAASBAMAAADYho+9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfQ6UgAAAAABJRU5ErkJggg==\n",
       "prompt_number": 34,
       "text": "sin(a\u22c5t + b)"
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 35,
       "text": "\n\u239ba\u22c5cos(b) + p\u22c5sin(b)     \u2502                 \u239b          2      \u239e\u2502    \u239e\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 = 0\u239f\n\u239c       2    2                                                     \u239f\n\u239d      a  + p                                                      \u23a0"
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABhEAAAAlBAMAAAC+HdUxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 36,
       "text": "\n   \u239b 2\u22c5t\u22c5im(a)                          2\u22c5t\u22c5im(a)                          2\u22c5t\n-\u2148\u22c5\u239d\u212f         \u22c5sin(b)\u22c5sin(t\u22c5re(a)) + \u2148\u22c5\u212f         \u22c5sin(b)\u22c5cos(t\u22c5re(a)) + \u2148\u22c5\u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n\u22c5im(a)                        2\u22c5t\u22c5im(a)                                       \n      \u22c5sin(t\u22c5re(a))\u22c5cos(b) - \u212f         \u22c5cos(b)\u22c5cos(t\u22c5re(a)) - sin(b)\u22c5sin(t\u22c5re(\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                2                             \n\n                                                                         \u239e  -t\na)) + \u2148\u22c5sin(b)\u22c5cos(t\u22c5re(a)) + \u2148\u22c5sin(t\u22c5re(a))\u22c5cos(b) + cos(b)\u22c5cos(t\u22c5re(a))\u23a0\u22c5\u212f  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n\u22c5im(a)             \n      \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                   "
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simplify(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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       "prompt_number": 37,
       "text": "\n\u239b     2\u22c5t\u22c5im(a)                        2\u22c5t\u22c5im(a)                        2\u22c5t\u22c5im\n\u239c  \u2148\u22c5\u212f         \u22c5sin(b)\u22c5sin(t\u22c5re(a))   \u212f         \u22c5sin(b)\u22c5cos(t\u22c5re(a))   \u212f      \n\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239d                 2                                 2                         \n\n(a)                          2\u22c5t\u22c5im(a)                                        \n   \u22c5sin(t\u22c5re(a))\u22c5cos(b)   \u2148\u22c5\u212f         \u22c5cos(b)\u22c5cos(t\u22c5re(a))   \u2148\u22c5sin(b)\u22c5sin(t\u22c5re\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n       2                                 2                             2      \n\n                                                                        \u239e     \n(a))   sin(b)\u22c5cos(t\u22c5re(a))   sin(t\u22c5re(a))\u22c5cos(b)   \u2148\u22c5cos(b)\u22c5cos(t\u22c5re(a))\u239f  -t\u22c5\n\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u212f   \n                2                     2                      2          \u23a0     \n\n                  \nim(a)             \n     \u22c5Heaviside(t)\n                  "
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "source": "10"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(cos(sqrt(b**2-a**2)*t)-a/sqrt(b**2-a**2)*sin(sqrt(b**2-a**2)))*exp(-a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
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       "prompt_number": 38,
       "text": "\n\u239b       \u239b   ___________\u239e                        \u239e      \n\u239c       \u239c  \u2571    2    2 \u239f      \u239b     ___________\u239e\u239f      \n\u239c  a\u22c5sin\u239d\u2572\u2571  - a  + b  \u23a0      \u239c    \u2571    2    2 \u239f\u239f  -a\u22c5t\n\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + cos\u239dt\u22c5\u2572\u2571  - a  + b  \u23a0\u239f\u22c5\u212f    \n\u239c         ___________                           \u239f      \n\u239c        \u2571    2    2                            \u239f      \n\u239d      \u2572\u2571  - a  + b                             \u23a0      "
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 39,
       "text": "\n\u239b      \u239b   ___________\u239e                                                       \n\u239c      \u239c  \u2571    2    2 \u239f                                       \u2502               \n\u239c   sin\u239d\u2572\u2571  - a  + b  \u23a0                   1                   \u2502               \n\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argume\n\u239c             ___________             \u239b      \u239b   2    2\u239e\u239e     \u2502               \n\u239c  \u239b    p\u239e   \u2571    2    2              \u239c      \u239c  a    b \u239f\u239f                     \n\u239c  \u239c1 + \u2500\u239f\u22c5\u2572\u2571  - a  + b               \u239c    4\u22c5\u239c- \u2500\u2500 + \u2500\u2500\u239f\u239f                     \n\u239c  \u239d    a\u23a0                    \u239b    p\u239e \u239c      \u239d  4    4 \u23a0\u239f                     \n\u239c                           a\u22c5\u239c1 + \u2500\u239f\u22c5\u239c1 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                     \n\u239c                             \u239d    a\u23a0 \u239c               2 \u239f                     \n\u239c                                     \u239c      2 \u239b    p\u239e  \u239f                     \n\u239c                                     \u239c     a \u22c5\u239c1 + \u2500\u239f  \u239f                     \n\u239d                                     \u239d        \u239d    a\u23a0  \u23a0                     \n\n                                                                              \n  \u239b                         2          \u239e\u2502                                     \n  \u239c          2               \u239b    p\u239e   \u239f\u2502       \u239b                            \u03c0\nnt\u239cpolar_lift (a)\u22c5polar_lift \u239c1 + \u2500\u239f, \u221e\u239f\u2502 < \u03c0 \u2227 \u239c\u2502periodic_argument(a, \u221e)\u2502 < \u2500\n  \u239d                          \u239d    a\u23a0   \u23a0\u2502       \u239d                            2\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                                                         \u239e\n                                    \u2502                 \u239b   2    2   \u239e\u2502    \u239f\n                               \u03c0\u239e   \u2502                 \u239c  a    b    \u239f\u2502    \u239f\n \u2228 \u2502periodic_argument(a, \u221e)\u2502 \u2264 \u2500\u239f \u2227 \u2502periodic_argument\u239c- \u2500\u2500 + \u2500\u2500, \u221e\u239f\u2502 = 0\u239f\n                               2\u23a0   \u2502                 \u239d  4    4    \u23a0\u2502    \u239f\n                                                                         \u239f\n                                                                         \u239f\n                                                                         \u239f\n                                                                         \u239f\n                                                                         \u239f\n                                                                         \u239f\n                                                                         \u239f\n                                                                         \u23a0"
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "p/(p**2+2*a*p+b**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAAlBAMAAABYAWEdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHaZEGZEMondu+8i\nq80NNKsLAAACBElEQVQ4Ec2Uz0sbQRTHv8ludpofklDofbE9FLEQ1IOIh6UgXlpSsaHRg0TrRQSJ\nuXpIDkLpybR/gZdc1lOhtD30kEvPij148C/ooRisSUEC6+y+TWZZZ0bw5DvMvPf9vO8w2QwPUETi\n6cKyo2BqeaFkG0U1VpDNBh61FEwjTyBzqMEK1Ea2qUAaeYBtR4PliP3HJznRqal2pabjcjZmy3W9\nmq3quZweO3Jdq1oXM1r+4KF3n3g4v4qNn926TP3FXS/tJfacmM202deYFC3ZRQ0N5KtRjeesiz7f\nKnOFGAjLBm9BqROHNXMAJDusFwO7VK/52yzloRQURhPI2jgnMlqfUPabb6ki5aEUFMt8TTdx4BAa\nrmHLF15vgFjEZdlm0HgEjC+18HimTO+cWnK9is2e/aLPTFIwWeuv3vku8xIZ22iZby8xHZxCLak+\nvh97XqCAJH+yznte19cyRazAKFqLXXyMuMZO8JksfCVXdLI+B34gXUDyEC5guO4f1z3hn6mAHrlG\nEsRkTdhmro+Sg8wb7vaDDi51LP7HhEESxGTdQJJdYSWBfMeitxK6wO84jNA1mqysXZ7Az9zfJBo1\nfk0/qCWNLScohSQma9bzrvG+/HoSq+UP1EUuNrU/MoUHySbr6bBpfZiInSTZZP0mmhSZZLLmrhS9\nQpZM1p1/juDSTEzWG4M5nt2wQRdRAAAAAElFTkSuQmCC\n",
       "prompt_number": 40,
       "text": "\n       p       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         2    2\n2\u22c5a\u22c5p + b  + p "
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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FjtNdKM5PBUVVepTbhapoIySpojwqrdHqxsiN8FzBz4cnjp/k1TCW077m4Jes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\nYmZZZXts7apM4xrrzEyyHW07ZlxmNq+ZTRZLGbT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       "prompt_number": 41,
       "text": "\n\u239b                    \u239b        \u239b    ___________________________________________\n\u239c                    \u239c        \u239c   \u2571                         2                 \n\u239c                    \u239c        \u239c4 \u2571  \u239b                  \u239b 2\u239e\u239e    \u239b  2        \u239b \n\u239c\u239b        _________\u239e \u239c     -t\u22c5\u239c\u2572\u2571   \u239d2\u22c5re(a)\u22c5im(a) - im\u239db \u23a0\u23a0  + \u239dre (a) - re\u239db\n\u239c\u239c       \u2571  2    2 \u239f \u239c        \u239d                                               \n\u239c\u239d-a - \u2572\u2571  a  - b  \u23a0\u22c5\u239c- \u2148\u22c5\u212f                                                   \n\u239c                    \u239d                                                        \n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239c                                                                             \n\u239c                                                                             \n\u239d                                                                             \n\n______________                                                                \n            2     \u239b     \u239b                  \u239b 2\u239e    2        \u239b 2\u239e     2   \u239e\u239e   \n2\u239e     2   \u239e      \u239catan2\u239d2\u22c5re(a)\u22c5im(a) - im\u239db \u23a0, re (a) - re\u239db \u23a0 - im (a)\u23a0\u239f   \n \u23a0 - im (a)\u23a0  \u22c5cos\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + \n                  \u239d                           2                           \u23a0   \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n     \u239e                                                                        \n     \u239f                                                                        \n     \u239f    \u239b  \u239b    _________________________________________________________   \nre(a)\u239f    \u239c  \u239c   \u2571                         2                             2    \n     \u23a0    \u239c  \u239c4 \u2571  \u239b                  \u239b 2\u239e\u239e    \u239b  2        \u239b 2\u239e     2   \u239e     \n      \u22c5sin\u239ct\u22c5\u239c\u2572\u2571   \u239d2\u22c5re(a)\u22c5im(a) - im\u239db \u23a0\u23a0  + \u239dre (a) - re\u239db \u23a0 - im (a)\u23a0  \u22c5si\n          \u239d  \u239d                                                                \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                           \u239b  \n                                                                           \u239c  \n                                                                  \u239e\u239e       \u239c4 \n \u239b     \u239b                  \u239b 2\u239e    2        \u239b 2\u239e     2   \u239e\u239e        \u239f\u239f    -t\u22c5\u239c\u2572\u2571\n \u239catan2\u239d2\u22c5re(a)\u22c5im(a) - im\u239db \u23a0, re (a) - re\u239db \u23a0 - im (a)\u23a0\u239f        \u239f\u239f       \u239d  \nn\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + im(a)\u239f\u239f + \u212f      \n \u239d                           2                           \u23a0        \u23a0\u23a0          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                        \u239b       _________    \u239e\n                                                        \u239c      \u2571  2    2     \u239f\n                                                       \u0393\u239d- 2\u22c5\u2572\u2571  a  - b   + 1\u23a0\n\n  _________________________________________________________                   \n \u2571                         2                             2     \u239b     \u239b        \n\u2571  \u239b                  \u239b 2\u239e\u239e    \u239b  2        \u239b 2\u239e     2   \u239e      \u239catan2\u239d2\u22c5re(a)\u22c5\n   \u239d2\u22c5re(a)\u22c5im(a) - im\u239db \u23a0\u23a0  + \u239dre (a) - re\u239db \u23a0 - im (a)\u23a0  \u22c5cos\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                               \u239d              \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                  \u239e                           \n          \u239b 2\u239e    2        \u239b 2\u239e     2   \u239e\u239e        \u239f                           \nim(a) - im\u239db \u23a0, re (a) - re\u239db \u23a0 - im (a)\u23a0\u239f        \u239f    \u239b  \u239b    _______________\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + re(a)\u239f    \u239c  \u239c   \u2571               \n             2                           \u23a0        \u23a0    \u239c  \u239c4 \u2571  \u239b             \n                                                   \u22c5cos\u239ct\u22c5\u239c\u2572\u2571   \u239d2\u22c5re(a)\u22c5im(a)\n                                                       \u239d  \u239d                   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n__________________________________________                                    \n          2                             2     \u239b     \u239b                  \u239b 2\u239e   \n     \u239b 2\u239e\u239e    \u239b  2        \u239b 2\u239e     2   \u239e      \u239catan2\u239d2\u22c5re(a)\u22c5im(a) - im\u239db \u23a0, r\n - im\u239db \u23a0\u23a0  + \u239dre (a) - re\u239db \u23a0 - im (a)\u23a0  \u22c5sin\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                              \u239d                           2   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                   \u239e                                          \n                                   \u239f                                          \n                                 \u239e\u239e\u239f                                          \n 2        \u239b 2\u239e     2   \u239e\u239e        \u239f\u239f\u239f  \u239b      _________\u239e                       \ne (a) - re\u239db \u23a0 - im (a)\u23a0\u239f        \u239f\u239f\u239f  \u239c     \u2571  2    2 \u239f   \u239b        _________\u239e \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + im(a)\u239f\u239f\u239f\u22c5\u0393\u239d-2\u22c5\u2572\u2571  a  - b  \u23a0   \u239c       \u2571  2    2 \u239f \n                        \u23a0        \u23a0\u23a0\u23a0                      \u239d-a + \u2572\u2571  a  - b  \u23a0\u22c5\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n\u239b        \u239b          \u239b   _________\u239e\u239e                                         \u239b \n\u239c        \u239c          \u239c  \u2571  2    2 \u239f\u239f    \u239b  \u239b          \u239b   _________\u239e\u239e\u239e       \u239c \n\u239c     -t\u22c5\u239dre(a) - re\u239d\u2572\u2571  a  - b  \u23a0\u23a0    \u239c  \u239c          \u239c  \u2571  2    2 \u239f\u239f\u239f    -t\u22c5\u239dr\n\u239d- \u2148\u22c5\u212f                             \u22c5sin\u239dt\u22c5\u239dim(a) - im\u239d\u2572\u2571  a  - b  \u23a0\u23a0\u23a0 + \u212f     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                          \u239b     _________    \u239e\n                                                          \u239c    \u2571  2    2     \u239f\n                                                         \u0393\u239d2\u22c5\u2572\u2571  a  - b   + 1\u23a0\n\n                                                                             \u239e\n                                                                             \u239f\n                                                                             \u239f\n         \u239b   _________\u239e\u239e                                  \u239e                  \u239f\n         \u239c  \u2571  2    2 \u239f\u239f    \u239b  \u239b          \u239b   _________\u239e\u239e\u239e\u239f  \u239b     _________\u239e\u239f\ne(a) - re\u239d\u2572\u2571  a  - b  \u23a0\u23a0    \u239c  \u239c          \u239c  \u2571  2    2 \u239f\u239f\u239f\u239f  \u239c    \u2571  2    2 \u239f\u239f\n                        \u22c5cos\u239dt\u22c5\u239dim(a) - im\u239d\u2572\u2571  a  - b  \u23a0\u23a0\u23a0\u23a0\u22c5\u0393\u239d2\u22c5\u2572\u2571  a  - b  \u23a0\u239f\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n                                                                             \u239f\n                                                                             \u239f\n                                                                             \u23a0\n\n             \n             \n             \n             \n             \n             \n             \n\u22c5Heaviside(t)\n             \n             \n             "
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "markdown",
     "source": "11"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "cos(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAASBAMAAAATXWtDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVJl2Zs0i7zIQRN2r\nibuk4LwrAAABKklEQVQoFWWQMUvDQBTHf6Fpampqi7rZQVoXcQm2S51SFaGTXcUliwg6ZHW8QVcb\nXFyEBr+AcXUKODkZcBMEwS/gJLqIr7G5Yn1wd7/7/9893j2YCk/fJ6Qlx9d4oUlDd0yWwnC1mkM8\nhrkU9nIxP3V+NYJ2ruanI2IWA9nPZM23PBa2PLqLfaiKUGgvcb3fzC6FvnVSWealnPIAQ3HfWBtt\nUAoZ1uzVwCc4uiOCQzBjGvAp7qzPjYJdqeI9fgsdy4uUW2beR67Sbseqf8AlBJH9hfkqblYZK1AE\np4qnrJFBaMY1xzV+u4rZqSQ0zIQNKCpK/kHiFms9OJf39VbE5rpntuRXFFzKK71732gq2Bb3T9j5\nJEfqv0nyPEmWCUyHE2rlSpMG29XYgR8bajv4246+AgAAAABJRU5ErkJggg==\n",
       "prompt_number": 42,
       "text": "cos(a\u22c5t)"
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 43,
       "text": "\n\u239b   p        \u2502                 \u239b          2      \u239e\u2502    \u239e\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 = 0\u239f\n\u239c 2    2                                               \u239f\n\u239da  + p                                                \u23a0"
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAo4AAAAlBAMAAAA+ZHXXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIrt2EKtEMolU3Zlm\n781wDQkSAAAIQklEQVRoBe1Zb4wcZRn/ze3s7L/Zu4kYpIa6Z5VQiolDt1Ajpx3BikrNLn8NkniT\nCFIsppM2NSSit6ghRdKwSdMYW+KuX1oIJK4cJtU7dHEL2N6WXtNG/EDCfQBN/cLe5ezFYDifd975\nv3N3s7cZc9i+yb3zvM/7PL/f8zzzzvvO7AF2267b0uVrHxV4dFu1D+/Lrk4Fhi7X0alFP4Jbx0o/\nMB8SX1GNK1CnjgOXxMIsxl7H43ExrCnc2FaLvR6lmTWVb1zBiOdjQrbrmNW7Cb6rdet600gP9GYf\nv/Wr3RT9p0mYdh2/6MXfe6OB2/DYl6/3KnuSU8VvQjRSJ25p9OQWv3Fe8XP0l6aLZdfxr64Kopo6\nj0moGCgHWD1Gy4u34kdarpkArhGWN/xfz6YDN7a/NN3orTpKHVeFVAcX8Ws0gRHdo+5FHMOQnlVk\nIPFaL27x2wreRImuvzSdeK/67PM6GySHWW83Q5zdeU5/BThpJIqftLXOlcrjb03/kEYplK48cJOs\nQP4FiptDVnUEjC5Qdmd9rQtk+dkmm37LZ7Ncmj5D/8AkDmPPqj7DXCVTRuZEpq6Pi4EbSHZ3MFux\nZcgHLK/bfc58cByHgGe0w1JOLYVMR8II+CXN+7FjK/5yjs+IjYCFbxhO8TufzXJp+gxpIN2p2SoO\nHcJeMkO0zTCFXJMGwllhPjVDAjXZeWOQq8DHkMOxO5LzfC6j8qunlxu0w1JbrxS0H3v0lhgJw7R1\neXGSIjLQyO5TRjWOc9CCC7ssQXHKtp2amJho+tJMvTynPvr2RtuAX09r7Gr2u+0icWiEsNe4k9VL\nqpg1NAr7VKaeqHBl5h3bJEvCRqrj05rwvqVz5mwbfAfaMZH45PphfM7ROkIkDNPa5UUdGKigkZ5H\nocGRBjV+DeuXoNjts/WnSciUlc8AvzLHZu8sNg6NEPYv+Jz3fun1IcmQj+L7Ujn7iG+KBuw2XERa\nPwP8ybpDI0Gb1Jarjc5DOKVL0w/iaDM4HQkj6JShVT+k4wm5g9wsn0x0IzteS4RZMhwLEvxphtXR\nY+3UkUMjhP3nHnMIi4udHK3G7ePXYrL1e+8Uk++lY6RDu2RNQWmYT3Y9uYXFRUxWId6/2ci1b65y\nK08fBcNjbopJHRijI3HdLMQ5Ppma4dewfgmKIV8w/jSj1tGCDmH/thtJa0pxB5Z08pmZPcek966e\nUknRgfTH51V8tWbgQR3Fk3W6rT4PeeS3vrE5OFJs4mCrCYKi8YoYHMDHSyS7/jkO8ckFSPciMXLY\nxOGGZh+JYqjJbMOSBKw6Jij+Ha2X5N+U03O3bTLkX76R2mRgV5sWTtEsjhU9ZRFsGxzFQEVgxfI1\nqYK7xQWUVIE2EHaAD5BJunb/Jp2kXB3Ia177jbjCOzTlxIzwJr0D72dQUTBMJx8vSqT7FP09+a8X\njzLpDySfN+14F40iz7ILS5LU6ff3739PwwTWay/iDdAW0KBuJ5JayZBncEbJNQRitCvgsg8tUqO6\n4F0eCvUHkFacgSVIR6FjAXQrF+jInKb1ZyDxg5rxLTqJkGsAWa9LehqPBxFQqoovFRQUrmRQK2Nw\nfx8v1pOS+Lc+vJCs0/lF6bLXaLdFo8hWyCMsSVLz9ZhaQLb5CAraQEMyKNXv6RL1pSoKypiBF5zo\nGftHNlBjN9dq19D1Htb0uRa9fwfb6JzB6mjQ+QKhbm5TUrWmvAz8Ddkq1dEgPYPcoFKRPzPiLSvH\n2k0WZ2iuyaCWxLDDSrJQaBfy8uLjfF9+ODObm0a+gucI+ZiHNxpFniJEaJJ2HeX/tKeqYmubIs2n\n2ZJJLl5LPb30FJQz7TYdF1YFTHaendPfaUsCVaqricKhWaeO4jTwZ2ZS054GmZe0wHNd0NlsoNlJ\nbmFQWBGDe/t48Q16iR2micx8uoOCLtIe4z5ZJEejoIeKRR3a+HqUOzR5yqDT+YUdrI7aulHFrqPp\n5kTv3VU4oFNHkbLUuM7tZQM/derIdte7UBVwGtuQmqd3TCJzbQEKQPSOTZkeC6FgoPAYg2LnzPIY\n3N/PSyRJNQNFnk50MKbJ9aqJ41BFoyArhCZJOLyOrMrGv1FYh89vYXV8HWmVnmud1iMtkZQTvZ+d\nh+G+91wHWec6t5fLuInVUWfPNf4BXJC1JJ6iOipnhXcTwB7XljbxKl7zjk05Uccr6TImTKiVMbi/\nn3fQwGD1BMrpRqIj5JVPlFWkhj1E0SjMpyUsSUKyzuufQP7oRdSuQKnB6vgqEvqQkRjGNiM9DLbr\n8Qr42Xkg7vfM7e0buMrTy8VWc98HD41e2PdBEygCR7ZC+NmnUTiu7GpvJ/u2xxjiZEvxjrl8qKhj\n6uYmgyLFShjcx8+bUJEZN3DVLfQd9bg0eeIu+hWEYTktEsWYRvZhSdLuS9+FP6TvQnl8Kw6N7L0P\nSQOp0b+3Rkaox5F27QKmiga58+i97PJT15lh0FEauX2dW35Fg1i1nN6J7GwZrgZDrJvO4gPA1yyY\nQc0SQi5LUJwOMe1VZUF72G/AWJOh5O2KRIBM60Ejes/rsa0K44kukp1dGlexBMU9rsWqJQvaw34O\nyWkGl2z0AHp30HZ7ULHyeDUYSS2AK5YDCt8wnOItn80qBya0l/1Zc2+l39bqPSDeGrTdElSsPF4N\nRtd/8RPackShFOyLrP9mQvvZB8z1iDf7B/9QINCxG08rqSbujfGgrznUvB5TSNdz3LwSE/4ag90c\nUzyJCgc2v7hi4lhLsGdjCuag/RV4X0wEaws22YwnHkmlf/ebbfCSeLDjOgb2tFvDvI7etyGu+T/s\nhUZMSb29uDgcE/SlAPtf3EjBHRzgbdAAAAAASUVORK5CYII=\n",
       "prompt_number": 44,
       "text": "\n \u239b   2\u22c5t\u22c5im(a)                 2\u22c5t\u22c5im(a)                                      \n-\u239d\u2148\u22c5\u212f         \u22c5sin(t\u22c5re(a)) - \u212f         \u22c5cos(t\u22c5re(a)) - \u2148\u22c5sin(t\u22c5re(a)) - cos(t\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                      2                       \n\n       \u239e  -t\u22c5im(a)             \n\u22c5re(a))\u23a0\u22c5\u212f        \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                               "
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "markdown",
     "source": "12"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "cos(a*t+b)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFcAAAASBAMAAADcc1+AAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6wAAAABJRU5ErkJggg==\n",
       "prompt_number": 45,
       "text": "cos(a\u22c5t + b)"
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 46,
       "text": "\n\u239b-a\u22c5sin(b) + p\u22c5cos(b)     \u2502                 \u239b          2      \u239e\u2502    \u239e\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 = 0\u239f\n\u239c       2    2                                                      \u239f\n\u239d      a  + p                                                       \u23a0"
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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OEtgkAyucgf7aCgMGcKcFHXZ+VR4k/SQDR28GzugStfS4DPycPEj6SQaO3gx069r9EBwp\n6GxdGiTdJANHbQayze5Qy1UhvUMAf++OmwQ1ycDKZuDMlYUTaDfPTg+KAXXkT7LyfNJPMnA0ZSA1\n2SU2by8tDXYJOoFNMvCJycB/AZbvnMpynTBIAAAAAElFTkSuQmCC\n",
       "prompt_number": 47,
       "text": "\n   \u239b     2\u22c5t\u22c5im(a)                        2\u22c5t\u22c5im(a)                        2\u22c5t\n-\u2148\u22c5\u239d- \u2148\u22c5\u212f         \u22c5sin(b)\u22c5sin(t\u22c5re(a)) + \u212f         \u22c5sin(b)\u22c5cos(t\u22c5re(a)) + \u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n\u22c5im(a)                          2\u22c5t\u22c5im(a)                                     \n      \u22c5sin(t\u22c5re(a))\u22c5cos(b) + \u2148\u22c5\u212f         \u22c5cos(b)\u22c5cos(t\u22c5re(a)) - \u2148\u22c5sin(b)\u22c5sin(t\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                 2                            \n\n                                                                           \u239e  \n\u22c5re(a)) - sin(b)\u22c5cos(t\u22c5re(a)) - sin(t\u22c5re(a))\u22c5cos(b) + \u2148\u22c5cos(b)\u22c5cos(t\u22c5re(a))\u23a0\u22c5\u212f\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n-t\u22c5im(a)             \n        \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                     "
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "markdown",
     "source": "13"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sinh(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAASBAMAAAD2w64vAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZlTNRImrEN27MiJ2\n75lZwTvPAAABPElEQVQoFW2RPUvDQBjH/2njRWNLrIIFUSi6iJsg6BgQwUk6+AH8AEJuctMGP4Gj\niMOBIHRq0MWl9DL5OdwciigKQhT1uZemRvNA/ve8/J57LndAic3kubGnU9PCVFiSEyfWW+La6WYm\nnrVp6qhI468YApaILTHVBjatbxdDjPoQhEBaRjAqaOuRXgHO8px/LrY/GkMgU4KAvlq6ir0tiijY\nAeMd4X1iwPGlBB0iLvGoBKhz7IYuAoGXXNCnjWM80ZZETCZg3xeqmKlzacEddbZxCv9ZEQL8JkqK\nxDEwCL13OGdmygImZJGgw/W4E7eYrOiTzqMWEvGmpmhBVaCe3B7IaqsJdIHFNPWjjevXtWhdC/2p\nhPvQPEwqQwE0aNI/8+JfqT+3biv7Y8KlpykxRtdm7WjkFFdP5vE98APjglaRJJPX+wAAAABJRU5E\nrkJggg==\n",
       "prompt_number": 48,
       "text": "sinh(a\u22c5t)"
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 49,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a/(p**2-a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAlBAMAAADioul9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdmZU3buZ74kiRBAy\nzauYRQn6AAABV0lEQVQ4EWNgQAeCyg7oQhh8VgE2wopUGNgSMHSiC1QwsBigi6HzOX8x7N+ALojO\n5/nNoMKMLojBr+S8x4ghiC6wTGmfJLrYKJ+B4T9h8IFG4cQj+AbD5INyB1DFNjIcR08m3AI81aiK\nzjPwL0AVYeD5wPALVYiHYf8EVBEGhgPcP9GFMiACh5HE2RqQOCAmUwJEwAhCgUk1JDaYuZAB4nAk\nRVwC3AhVjEImDDzSkw6ARSCKmMU2qW04uFMHoSiQIYVh/v//EAGIok37Bdgy//9HxAqTA0MaQgdE\n0erzDBwOYEHOUBAQYG9gcIYqYgsNvRIa+gDIE2VgDYAKgqn5C7h/I/gQkxiCGIB6kcD5DUwOBnA+\nVNFPhuUb4GJABvuEuQ8E4AIQRTx/4E6AyHAlaT+cgKaIKUjjAFwIk7EYLMSHMBpTCUyEfQGMhYee\nj+Jq7Aq57mdDJQAooGWrg1O6jwAAAABJRU5ErkJggg==\n",
       "prompt_number": 50,
       "text": "\n    a    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   2    2\n- a  + p "
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAn4AAAAlBAMAAAAuKPlOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ncmXlbB9ml0yCEfgPzsjEEbDV3NEAAAAASUVORK5CYII=\n",
       "prompt_number": 51,
       "text": "\n\u239b   2\u22c5t\u22c5re(a)                 2\u22c5t\u22c5re(a)                                       \n\u239d\u2148\u22c5\u212f         \u22c5sin(t\u22c5im(a)) + \u212f         \u22c5cos(t\u22c5im(a)) + \u2148\u22c5sin(t\u22c5im(a)) - cos(t\u22c5\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                     2                        \n\n      \u239e  -t\u22c5re(a)             \nim(a))\u23a0\u22c5\u212f        \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                              "
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "markdown",
     "source": "14"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "cosh(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAASBAMAAADyNn4SAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVJl2Zs0i7zIQRN2r\nibuk4LwrAAABU0lEQVQoFW2RMUvDQBTHf6VpampqizoIdpDWRVyC7VKn1Bahkx1cxCWLCDpkdcyg\nm7TFxUVo8AsYV6eCk1MLboIg+AVEVKqD+NKkKdQ+uPfe/e/37t7dwTQzI3GcBZLqBFG3gij+IsqC\npDkIYiXUpSRmTDAh4oXyXB/2piJRZcaF0lREl5WhtcSfyZgvmixUTSqLDQa1gigZGfHSMtf7MpNJ\nvKGepFd4TvV54B27DR1BXln3HSTbdLLamm1hH93h8kHGgUNQPPLgdz9rcSPajuxn9n4d0fwOj6W2\nzy0zbz7iREhZzX2FyCXYrvaN8iLI8CBU28E+dXgMEemw1Va8rG7EgnY9ttNd8kqXTX6GByUcktZB\n10hk69CUnXJFl60NUylWzfPPpd6uXNIgtVq/t2IFB2qC/Ddt9AH+0uQHhPjTuEyea6rp8n6hXY2S\niagZkVCGP6E0TxkYcRiKAAAAAElFTkSuQmCC\n",
       "prompt_number": 52,
       "text": "cosh(a\u22c5t)"
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 53,
       "text": "\n\u239b\u23a7                      \u2148\u22c5\u03c0                         \u2502 2\u2502                      \n\u239c\u23aa                   p\u22c5\u212f                            \u2502p \u2502                      \n\u239c\u23aa                  \u2500\u2500\u2500\u2500\u2500\u2500\u2500                     for \u2502\u2500\u2500\u2502 < 1, Max(0, cos(Abs(p\n\u239c\u23aa                   2    2                         \u2502 2\u2502                      \n\u239c\u23aa                  a  - p                          \u2502a \u2502                      \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                  \u2502 2\u2502                      \n\u239c\u23aa                     p                            \u2502a \u2502                      \n\u239c\u23aa                 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                    for \u2502\u2500\u2500\u2502 < 1                  \n\u239c\u23aa                    2    2                        \u2502 2\u2502                      \n\u239c\u23aa                 - a  + p                         \u2502p \u2502                      \n\u239c\u23aa                                                                            \n\u239c\u23aa  \u23a7                                 \u2502 2\u2502                                    \n\u239c\u23aa  \u23aa            -p                   \u2502p \u2502                                    \n\u239c\u23aa\u03c0\u22c5\u23aa       \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500          for \u2502\u2500\u2500\u2502 < 1                                \n\u239c\u23aa  \u23aa           \u239b     2\u239e              \u2502 2\u2502                                    \n\u239c\u23aa  \u23aa           \u239c    p \u239f              \u2502a \u2502                                    \n\u239c\u23aa  \u23aa       \u03c0\u22c5a\u22c5\u239c1 - \u2500\u2500\u239f                                                      \n\u239c\u23a8  \u23aa           \u239c     2\u239f                                                      \n\u239c\u23aa  \u23aa           \u239d    a \u23a0                                                      \n\u239c\u23aa  \u23aa                                                                         \n\u239c\u23aa  \u23aa                                 \u2502 2\u2502                                    \n\u239c\u23aa  \u23aa            a                    \u2502a \u2502                                    \n\u239c\u23aa  \u23a8      \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500         for \u2502\u2500\u2500\u2502 < 1                                \n\u239c\u23aa  \u23aa          \u239b   2    \u239e             \u2502 2\u2502                                    \n\u239c\u23aa  \u23aa          \u239c  a     \u239f             \u2502p \u2502                                    \n\u239c\u23aa  \u23aa      \u03c0\u22c5p\u22c5\u239c- \u2500\u2500 + 1\u239f                                                     \n\u239c\u23aa  \u23aa          \u239c   2    \u239f                                                     \n\u239c\u23aa  \u23aa          \u239d  p     \u23a0                                                     \n\u239c\u23aa  \u23aa                                                                         \n\u239c\u23aa  \u23aa        \u239b             \u2502  2\u239e                                              \n\u239c\u23aa  \u23aa\u256d\u2500\u256e2, 1 \u239c 1/2    0, 0 \u2502 p \u239f                                              \n\u239c\u23aa  \u23aa\u2502\u2576\u2510     \u239c             \u2502 \u2500\u2500\u239f   otherwise                                  \n\u239c\u23aa  \u23aa\u2570\u2500\u256f3, 3 \u239c0, 1/2   0   \u2502  2\u239f                                              \n\u239c\u23aa  \u23a9        \u239d             \u2502 a \u23a0                                              \n\u239c\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   otherwise                    \n\u239d\u23a9                     a                                                      \n\n                                                                              \n                                                             \u2502                \neriodic_argument(polar_lift(a)**2, oo))/2)*sqrt(Abs(a**2))), \u2502periodic_argumen\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                             2    \u239e\n \u239b          2      \u239e\u2502       p     \u239f\nt\u239dpolar_lift (a), \u221e\u23a0\u2502 < \u03c0 \u2227 \u2500\u2500 \u2260 1\u239f\n                             2    \u239f\n                            a     \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u23a0"
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "p/(p**2-a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAlBAMAAADioul9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 54,
       "text": "\n    p    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   2    2\n- a  + p "
      }
     ],
     "prompt_number": 54
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAn4AAAAlBAMAAAAuKPlOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nOth8OwAAAABJRU5ErkJggg==\n",
       "prompt_number": 55,
       "text": "\n\u239b   2\u22c5t\u22c5re(a)                 2\u22c5t\u22c5re(a)                                       \n\u239d\u2148\u22c5\u212f         \u22c5sin(t\u22c5im(a)) + \u212f         \u22c5cos(t\u22c5im(a)) - \u2148\u22c5sin(t\u22c5im(a)) + cos(t\u22c5\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                     2                        \n\n      \u239e  -t\u22c5re(a)             \nim(a))\u23a0\u22c5\u212f        \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                              "
      }
     ],
     "prompt_number": 55
    },
    {
     "cell_type": "markdown",
     "source": "15"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "-((b-c)*exp(a*t)+(c-a)*exp(b*t)+(a-b)*exp(c*t))/((a-b)*(b-c)*(c-a))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATQAAAAtBAMAAADSAYpLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 56,
       "text": "\n            b\u22c5t            c\u22c5t            a\u22c5t\n- (-a + c)\u22c5\u212f    - (a - b)\u22c5\u212f    - (b - c)\u22c5\u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n           (-a + c)\u22c5(a - b)\u22c5(b - c)          "
      }
     ],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 57,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 57
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/((p-a)*(p-b)*(p-c))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANMAAAApBAMAAABdIAGVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXN55AAAAAElFTkSuQmCC\n",
       "prompt_number": 58,
       "text": "\n            1             \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n(-a + p)\u22c5(-b + p)\u22c5(-c + p)"
      }
     ],
     "prompt_number": 58
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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0A+cyg+964zzlqUkFskHKlZFZNzAIbfUumUyyAeEuv5lMIBssG8TkshIMuqjJZfmBNplk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\nQmCC\n",
       "prompt_number": 59,
       "text": "\n \u239b     t\u22c5re(b)                   t\u22c5re(b)                     t\u22c5re(c)          \n-\u239d\u2148\u22c5a\u22c5\u212f       \u22c5sin(t\u22c5im(b)) + a\u22c5\u212f       \u22c5cos(t\u22c5im(b)) - \u2148\u22c5a\u22c5\u212f       \u22c5sin(t\u22c5im(\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n         t\u22c5re(c)                     t\u22c5re(a)                   t\u22c5re(a)        \nc)) - a\u22c5\u212f       \u22c5cos(t\u22c5im(c)) - \u2148\u22c5b\u22c5\u212f       \u22c5sin(t\u22c5im(a)) - b\u22c5\u212f       \u22c5cos(t\u22c5i\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n             t\u22c5re(c)                   t\u22c5re(c)                     t\u22c5re(a)    \nm(a)) + \u2148\u22c5b\u22c5\u212f       \u22c5sin(t\u22c5im(c)) + b\u22c5\u212f       \u22c5cos(t\u22c5im(c)) + \u2148\u22c5c\u22c5\u212f       \u22c5sin\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n (a - b)\u22c5(a - c)\u22c5(b - c)                                                      \n\n               t\u22c5re(a)                     t\u22c5re(b)                   t\u22c5re(b)  \n(t\u22c5im(a)) + c\u22c5\u212f       \u22c5cos(t\u22c5im(a)) - \u2148\u22c5c\u22c5\u212f       \u22c5sin(t\u22c5im(b)) - c\u22c5\u212f       \u22c5c\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n           \u239e             \nos(t\u22c5im(b))\u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                         "
      }
     ],
     "prompt_number": 59
    },
    {
     "cell_type": "markdown",
     "source": "16"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(exp(a*t)-(1+(a-b)*t)*exp(b*t))/(a-b)**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMIAAAAvBAMAAABDFQheAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 60,
       "text": "\n                   b\u22c5t    a\u22c5t\n- (t\u22c5(a - b) + 1)\u22c5\u212f    + \u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                  2          \n           (a - b)           "
      }
     ],
     "prompt_number": 60
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 61,
       "text": "\n\u239b                     1                           \u239b\u2502                 \u239b \u2148\u22c5\u03c0    \n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u239c\u2502periodic_argument\u239d\u212f   \u22c5pol\n\u239c     2                2    2          2    3     \u239d                           \n\u239d- a\u22c5b  + 2\u22c5a\u22c5b\u22c5p - a\u22c5p  + b \u22c5p - 2\u22c5b\u22c5p  + p                                  \n\n             \u239e\u2502   \u03c0   \u2502                 \u239b \u2148\u22c5\u03c0                 \u239e\u2502   \u03c0\u239e   \u239b\u2502    \nar_lift(b), \u221e\u23a0\u2502 < \u2500 \u2228 \u2502periodic_argument\u239d\u212f   \u22c5polar_lift(b), \u221e\u23a0\u2502 \u2264 \u2500\u239f \u2227 \u239c\u2502peri\n                  2                                                2\u23a0   \u239d     \n                                                                              \n\n             \u239b \u2148\u22c5\u03c0                 \u239e\u2502   \u03c0   \u2502                 \u239b \u2148\u22c5\u03c0           \nodic_argument\u239d\u212f   \u22c5polar_lift(a), \u221e\u23a0\u2502 \u2264 \u2500 \u2228 \u2502periodic_argument\u239d\u212f   \u22c5polar_lift\n                                        2                                     \n                                                                              \n\n      \u239e\u2502   \u03c0\u239e\u239e\n(a), \u221e\u23a0\u2502 < \u2500\u239f\u239f\n           2\u23a0\u239f\n             \u23a0"
      }
     ],
     "prompt_number": 61
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABL8AAAAxBAMAAADJpxYfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJz+UJNRNKkwZCP+75uSHkpx4TP4PZtq18SlMdkkAAAAASUVORK5CYII=\n",
       "prompt_number": 62,
       "text": "\n \u239b       t\u22c5re(b)                     t\u22c5re(b)                       t\u22c5re(b)    \n-\u239d\u2148\u22c5a\u22c5t\u22c5\u212f       \u22c5sin(t\u22c5im(b)) + a\u22c5t\u22c5\u212f       \u22c5cos(t\u22c5im(b)) - \u2148\u22c5b\u22c5t\u22c5\u212f       \u22c5sin\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                 t\u22c5re(b)                   t\u22c5re(a)                 t\u22c5re(a)    \n(t\u22c5im(b)) - b\u22c5t\u22c5\u212f       \u22c5cos(t\u22c5im(b)) - \u2148\u22c5\u212f       \u22c5sin(t\u22c5im(a)) - \u212f       \u22c5cos\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                        2                                     \n                                 (a - b)                                      \n\n               t\u22c5re(b)                 t\u22c5re(b)             \u239e             \n(t\u22c5im(a)) + \u2148\u22c5\u212f       \u22c5sin(t\u22c5im(b)) + \u212f       \u22c5cos(t\u22c5im(b))\u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                         \n                                                                         "
      }
     ],
     "prompt_number": 62
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/((p-a)*(p-b)**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJQAAAArBAMAAABsjrxHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWBHCBkhcauYXY/EY0TRgrAj8AS5x2qTe3qqPAAAAAElFTkSuQmCC\n",
       "prompt_number": 63,
       "text": "\n        1         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                 2\n(-a + p)\u22c5(-b + p) "
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABL8AAAAxBAMAAADJpxYfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIrt2MqsQ3VSJRO9m\nmc1UIBGWAAAMg0lEQVR4Ae1cfYxcVRX/zc7Mm4++6Y6CikKYTf1AIiGDiyXgVl6ojZIAXSQQbWL6\niBCMQGZVQkyDcRQLJgJdRUmAiBMDkkagSyRE+cfB5aPKLF1YAhqKblQqGhNm26Yhm0o95977Zt67\n973Z97bzR2vfTfrufeeec+7v/s6Zd++86V5gpfJLXFJdSWfF/rKbe3pFpXgKQ3E1DCdHScxRmkuy\njm4eQ4EQL2qslXFMXcvJXrvzosXi42ZXAsk1WL/lJ2fjh0efqBiKqyE4UcQkICGgepTmytdRzSNh\naL8WwL+qm/eFWFXalfJM5h1rxuxrB0V28DZwtxfzv0KteX1AKG90s3aIjl80wJVmqnv2eRngRLfS\nnPacKGJ69/6GbqP7JN1B5jD0dYfeYAPmoZkYLnsQwkJrICBv2aY3aqxaDKiN+rEQy0K1/HQX9xcn\njb68eBxZnXEXyL3twJo3VHqCr+ORGkoz23uCfuNy0bxhdj3Vlx0Ebux39Vos90q0qxUAxXOCOHgY\niyLGg+Wv/UCEPISaAeYIYBAOwkjhjlWTwcbRoQ0i8KISlh7sx1fKX3C9O8FjcOLZutfJtVwvy/df\ngHsO4dnKWa/IzhdcWQO7hc4Hkf0NSTbTv1u9Hl99smjv/HWr1HDtyZfnHF+faNpN4GRkpvFpauS6\nQDGAQqqznLS46K7sJ4WYLysAGuCk5wKIhYf0PWIw/smqNH/B9dwEgAihQY1p7hlTHcAg5GGkUIdO\nBiIwhEbHg9ALbRQCLyqbSCH3+v76xb8706dKTTmquO5SZKg5BHOi4PrtRqbF3XZk1h943S20fiE7\n7+opTQGkMw/83AFepN61sivXpLZXJBR6/p3U2PAktpXHvA6vLlDjTFSAEj2l7EW62+d19WshD3dV\n7KtHAVJ+Bjjpj4RYeFhfEVOpN5R1BDOiV1GjNLkyzH19AQxSLmeZa/q0qGnwGoEhPDp6aH2+Awi8\nqBSqpFE6RAF3farUlKOKa4N1uEgPXk5I2fdkpa6jLdHYC/u2Q5txPe4K9MonDelM01p+8x34MvVm\n20KlKCvRxrKoKm3sblzTtWaKi1Lcv/Jnexklur721XZ+nuqJfqfXEvKVXPGnPBSQ8hLLifjIrYiH\nPSpiau5lHkqvDgIRUkWNp0F1tLlcCnoYpI0kJUAtdUTzGsQQHp3I0GoIvKhkKdahCSYhimsvwWRc\nvZyQGl/yKcpFjwQ3Wch0N+Mh/DnQC+RbQocH/XwVvEzmFulCmdcWlbjkuqIqOC5quW55ssLqgbJF\nLIycYLnvYk2TaiNgEPIVXeVbEYDIJxd2vqITxMLD7hQxt+N8vvOXfMsPRPQoanxa0eYIYpA2khQ/\ntSyP5jXf8mMIj05kaDUEXlTKMzRk2BOMxF7pJZjk0csJ2b3N08pO3I5H/7kAzM5Vuw+Wyos1bLJu\nphXsBy/mznAeW+pQF0YhdB4BrL31zOE/1AGZTR4L47unyn/5dh32xEujZWf6sUw3N3mdSwm4MIHi\nxAR7IzddsFK5Tbv7WdQ6dwvHJA8Ukoe62n3H4kU3ld8+bY4GjwakXEU5WQUeHzEPYVt7EDNgMj1q\nFBLDHHeOt3HrLD3n71gUyj5OQDx6pHjUCj86r8mio4U2TlQsWh29BMtOvIQbZl+1vzNZ2n/NGY4c\nG492dlQxPlftzSE48SWBmy4fxmt8oS1Wpr63aV/cKnQqG8hqJ/Juw7EOoOYCDaGI3DhGxk6zl/EE\nIPfbioWRemUKIxT5M3FSZTeeuQ+3tBdeIqPP4FTnZyhNszdYXRqGlDbWsS7T3tNa42ANOdcKy0Nc\nladxlXUQjXqGpx4JSDmLcLIaPH5iOhuag5iRZCpqFBLDPLuYeaI0j3t5QvSBDXAieFSkBBJM5zVZ\ndLTQxmLhKcJf+u+9977t4g2c6r5MO++GQ7vwhiPHps3anmplPkN54M3BP3ERJabAniJ70J77fpSq\ndP9ZWC2W4+FWGaMODvMFp5KAdKisHUN+Et/kL81cFAvPozLPiqUZ/IikOQfyTZ31Dm68egmZJfaG\nzAx9Oh02w9+AK1Goo8BjBgvLQ1yVt6FFGEbbAkgkIOUrwslq8OjEDGBGkqmoUUgM80bTerVWRe1D\nPCGNE8GjIsWfYDqvAzCEREeHEIuFWwi/XCJzB1FoP0IPmpH5skMBlJnRaKJW3eyA9HpxlTkhJk77\nk/yVVLasmca3xG5l/2zw1538kY9wNlBIiYYP9HY0xWXe3xymbQn5eXnd2d9Y91Fq/BaFJr+9GD1r\nIpAxZcpKuwscZm/ITIm9AgPY0aKdHCVLwSHxOi51SjkuLZaHuMLW/QKNI75LhAPqeRLOQ5ysBo9O\nDAYww2RKat7Lc6KFAYb5LpryHqKqzRPSOBE8ClI8amWUdF4HYTCjo0OIxcLfCbtMMPtwZ65pzV5Y\nLR+i7fOoI+e/iz4j1T2dzp98c+CcUIUSTJZai1ZbewyZZa9L1e4pW6u9BLue8mSMXj65wF/dRrW8\nFFwiybjh4vdAraWsVeVN5QB7gzUDVrJvo69Eh6wlEEZziRTyEFdWZvsSp7tMsEhAcuAoJ6vAYxCD\naGYEmR41EolpvsuRCbaeJxTkRPIYskTqvA7AEBodFQ5VxWLBl2CcK887jSpuuYETTM5fJpjwqOIa\nmHhvidzs2lPNfL1IIYGrAIjqGZR4pZJPsFH6Ilkvwj5Azx6nwYskZIbK53juEJ4v4go0CYTld0IP\nY7znIDIH2ZuwISXsqaN4kBbuKwmuX1u0hTzEle3g7n6CRQKS/qKcrAaPTgyimRFketQoJIY5rS2Z\nmoPaJTyhICeSR0WKf4nUeR2AwYyOEdpYLPCjWD7B+DPivIPaKThvPSeYnH+jRU8wSvxcfw4qJ8TE\neVBR1lRPn6yvbT6Hj8NuKZmonkW2JR4XvESudcA6OH0Sn6DXag+7yI0JLcXC45k3s3jLdkea+KOQ\ne5fzYFd/Shs09gb8A6yEzJsYmcFU+WbgIk+xX7M8xJU9iQtkutN8owEpPxFOsAo8OjGIZkaQ6VGj\nkBjm2Sk8XZrEG2JCGieCR0WKP8EMXqMxhETHgBCHhacIv3pN8WPY71/GjpPQmOcEk2Nnx3ChUxoD\nb6xUXFVOyImzPZfy3ueuqBYXHNzYOUdK1HV2YiK39d+b3v3+1v/Qa1V65pAO8MADlyO3cA+tc22h\np1h4tHPtObjzXHqJMVsVcu9SvOIC2At7Qd5INA5WIuD/erOKS893gY6n2K9ZHuLKHp9tb3r3wa1v\nbXq3PQCQ8hPhBKvAoxPDc4lgRpDpUaOQGObYPt7C3IY2T4h0ApwIHhUp/gQzeI3GEBIdA0IcFugJ\nxD8VXUo/FdkL52L7xMYvIu+AZi7Hxp2dHW9hbtzpz0HlBAmo8P4zfrGmdN21rpCUm3rHoPvrjM59\nhiSeIBJQPHOlNQQ8JhAoamIiMTGo38/iU2tiSBQdEwFFJTcTEz+rKQ+BiT+bwJ5U79PVd+qCOPel\nlqZF74JWWYYCaBh4DCD0BjFJMTAgOSkGhkQQDAQMgBbw+EV5CIxKX8OSlLwb1LaSjN83varfFK1r\ntfv4t8MBNAQ8OhAkpUbHgOSk6BgSQtARMIC1TvxYgN8Y0xfiQE5UmkkcwKoH1bNu8D7m3Vc0vfXa\nffzb4QAaAh4dCLJu/Fmwpo4ByUnRMWRddhy76AgYwOdiW7Oi8JB1udkrY71W2kgZMBh4xZAkFQQW\nzKTGqf7/OQPJ94EGIUXHEKWClAHFgL5qpsSkDKQMpAycAAyoH/lHj6QlZWD4DHRPgI9QOsWUgeOX\ngY2fco5f8CnyY54Bq5578pgHmQI8fhmg/6a5fPyiT5Ef+ww4vf9Hd+xjTREeuwzYodDaJDX/1jNU\nNRWmDAxi4HKj0zsTZM7oSQUpA0kZsJuGBe2++KSWct0yulJBykBCBgqmvr1Isn3YePUzZl8qSRlI\nxsCtpnp+nmQTmSNH0nfVJjmpJCEDW0z9NU2SXWbKU0nKQHIG+CklTiHpm0ad1NLXSFspAzEZoL/a\nhTyFpG8QdVJLXyNtpQzEZCAzQ38pz6e5+ErUSS0+lbSZMhCPAT6jRZ5CEjxMRRxKEs9FqpUyEM2A\nNcNHwTRcnwb9QkSHfZgntfhU0mbKQFwGulCnkPQMxGEqISe19BTSRspAfAboLA95CknfJOqklr5G\n2koZiMsAnUciTyHpG0Sd1NLXSFspA3EZME8DkZb74jpI9VIGBjGgzvLQVYbwx6i6y/T+xGTgqtBp\nJz+UJNRNKkwZCP+75uSHkpx4TP4PZtq18SlMdkkAAAAASUVORK5CYII=\n",
       "prompt_number": 64,
       "text": "\n \u239b       t\u22c5re(b)                     t\u22c5re(b)                       t\u22c5re(b)    \n-\u239d\u2148\u22c5a\u22c5t\u22c5\u212f       \u22c5sin(t\u22c5im(b)) + a\u22c5t\u22c5\u212f       \u22c5cos(t\u22c5im(b)) - \u2148\u22c5b\u22c5t\u22c5\u212f       \u22c5sin\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                 t\u22c5re(b)                   t\u22c5re(a)                 t\u22c5re(a)    \n(t\u22c5im(b)) - b\u22c5t\u22c5\u212f       \u22c5cos(t\u22c5im(b)) - \u2148\u22c5\u212f       \u22c5sin(t\u22c5im(a)) - \u212f       \u22c5cos\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                        2                                     \n                                 (a - b)                                      \n\n               t\u22c5re(b)                 t\u22c5re(b)             \u239e             \n(t\u22c5im(a)) + \u2148\u22c5\u212f       \u22c5sin(t\u22c5im(b)) + \u212f       \u22c5cos(t\u22c5im(b))\u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                         \n                                                                         "
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "markdown",
     "source": "17"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(a*exp(a*t)-(a+b*(a-b)*t)*exp(b*t))/(a-b)**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMMAAAAvBAMAAACs12NgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 65,
       "text": "\n   a\u22c5t                      b\u22c5t\na\u22c5\u212f    - (a + b\u22c5t\u22c5(a - b))\u22c5\u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                   2           \n            (a - b)            "
      }
     ],
     "prompt_number": 65
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 66,
       "text": "\n\u239b                     p                           \u239b\u2502                 \u239b \u2148\u22c5\u03c0    \n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u239c\u2502periodic_argument\u239d\u212f   \u22c5pol\n\u239c     2                2    2          2    3     \u239d                           \n\u239d- a\u22c5b  + 2\u22c5a\u22c5b\u22c5p - a\u22c5p  + b \u22c5p - 2\u22c5b\u22c5p  + p                                  \n\n             \u239e\u2502   \u03c0   \u2502                 \u239b \u2148\u22c5\u03c0                 \u239e\u2502   \u03c0\u239e   \u239b\u2502    \nar_lift(a), \u221e\u23a0\u2502 \u2264 \u2500 \u2228 \u2502periodic_argument\u239d\u212f   \u22c5polar_lift(a), \u221e\u23a0\u2502 < \u2500\u239f \u2227 \u239c\u2502peri\n                  2                                                2\u23a0   \u239d     \n                                                                              \n\n             \u239b \u2148\u22c5\u03c0                 \u239e\u2502   \u03c0   \u2502                 \u239b \u2148\u22c5\u03c0           \nodic_argument\u239d\u212f   \u22c5polar_lift(b), \u221e\u23a0\u2502 < \u2500 \u2228 \u2502periodic_argument\u239d\u212f   \u22c5polar_lift\n                                        2                                     \n                                                                              \n\n      \u239e\u2502   \u03c0\u239e\u239e\n(b), \u221e\u23a0\u2502 \u2264 \u2500\u239f\u239f\n           2\u23a0\u239f\n             \u23a0"
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABQUAAAAxBAMAAACR/dIvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/gcdsChUXijduAAAAABJRU5ErkJggg==\n",
       "prompt_number": 67,
       "text": "\n \u239b         t\u22c5re(b)                       t\u22c5re(b)                     t\u22c5re(a)  \n-\u239d\u2148\u22c5a\u22c5b\u22c5t\u22c5\u212f       \u22c5sin(t\u22c5im(b)) + a\u22c5b\u22c5t\u22c5\u212f       \u22c5cos(t\u22c5im(b)) - \u2148\u22c5a\u22c5\u212f       \u22c5s\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                 t\u22c5re(a)                     t\u22c5re(b)                   t\u22c5re(b)\nin(t\u22c5im(a)) - a\u22c5\u212f       \u22c5cos(t\u22c5im(a)) + \u2148\u22c5a\u22c5\u212f       \u22c5sin(t\u22c5im(b)) + a\u22c5\u212f       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                               2                              \n                                        (a - b)                               \n\n                   2    t\u22c5re(b)                 2    t\u22c5re(b)             \u239e    \n\u22c5cos(t\u22c5im(b)) - \u2148\u22c5b \u22c5t\u22c5\u212f       \u22c5sin(t\u22c5im(b)) - b \u22c5t\u22c5\u212f       \u22c5cos(t\u22c5im(b))\u23a0\u22c5Hea\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n         \nviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         \n         "
      }
     ],
     "prompt_number": 67
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "p/((p-a)*(p-b)**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJQAAAAnBAMAAAAbTHw8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 68,
       "text": "\n        p         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                 2\n(-a + p)\u22c5(-b + p) "
      }
     ],
     "prompt_number": 68
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABQUAAAAxBAMAAACR/dIvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/gcdsChUXijduAAAAABJRU5ErkJggg==\n",
       "prompt_number": 69,
       "text": "\n \u239b         t\u22c5re(b)                       t\u22c5re(b)                     t\u22c5re(a)  \n-\u239d\u2148\u22c5a\u22c5b\u22c5t\u22c5\u212f       \u22c5sin(t\u22c5im(b)) + a\u22c5b\u22c5t\u22c5\u212f       \u22c5cos(t\u22c5im(b)) - \u2148\u22c5a\u22c5\u212f       \u22c5s\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                 t\u22c5re(a)                     t\u22c5re(b)                   t\u22c5re(b)\nin(t\u22c5im(a)) - a\u22c5\u212f       \u22c5cos(t\u22c5im(a)) + \u2148\u22c5a\u22c5\u212f       \u22c5sin(t\u22c5im(b)) + a\u22c5\u212f       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                               2                              \n                                        (a - b)                               \n\n                   2    t\u22c5re(b)                 2    t\u22c5re(b)             \u239e    \n\u22c5cos(t\u22c5im(b)) - \u2148\u22c5b \u22c5t\u22c5\u212f       \u22c5sin(t\u22c5im(b)) - b \u22c5t\u22c5\u212f       \u22c5cos(t\u22c5im(b))\u23a0\u22c5Hea\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n         \nviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         \n         "
      }
     ],
     "prompt_number": 69
    },
    {
     "cell_type": "markdown",
     "source": "18"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(a**2*exp(a*t)-(2*a-b+b*(a-b)*t)*b*exp(b*t))/(a-b)**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAP0AAAAvBAMAAADa7dHrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdmZU3buZ74kiRBAy\nzauYRQn6AAAEMklEQVRYCe1XQYgbVRj+MplNmsyYzC60rFZkTFspVTGF4k0ZEYXaHuzCYi9KXKxU\nUFjpxaKH9STqwSiKIMqmoNKDaApiEaSZHtTTugWPSt1eWmqlxIKHboX0///3ZjJvJ9udQ6Z70Ad5\n73/f9///N++9yUt+gJs7+ZeM3DlhbOZpFI8DP0QCp/BzN7JLfmTlOjrzwFyksIx6J7LtdmQlRz85\nGWFbIzAgHRX7FVeA/VGQi55Sndz6yVt3RSiP7h8hD7WRD4Wlf5nkdlkNiZ657xJzlctpaag08/LJ\nVzsxf1SsktfDozEmxrL0O4DdR5omQzO3T92n9LFGc0XPCOFcezVid77gLdDNaomx2F0aHoqiHpQh\nQKHtXlMI967Ss86RfZg+NoNmE+43A+Ncd3QVdBqPJN60ExB4H6Zx0DFiLvKM1lH2cGlIFH2xay0a\nVukTrUpQ1Qk3lQAAzlXwqaP2NY7aoX4WuNs/CBX4lN//iq24HWOr1oG9gPcid3ogn1Cg3NRn8KdM\njY45LBkQ53LPKegX50C9GpJdmNqHxcGAznDqpVJjW3OO4+JWubbbA+oyfwaY3BGIqfUXG29WX3mS\nHPqERpw4AMRFgRpRudiVpba1S/RWAYfwguIP431lJHtrFW8APYacvzHhlQJhtf5DnXI4QfIOJY05\ncQCYQ7mrZzyoXK+TlZCyArzIJLbM420xjK62gseAjxmbaGEnSi2htf5V2F49BCrzGHLiADAH4zJR\nufabUuUFPC4R9Zmptg6lhLPcPDlheu0/ZGIaeA12k6wjs88+MfscGcdQ9/lLVQli7iMOPKQ52GEq\n10E6lYTUYse5Qd50Xh0Z1nS9dvU6cJrQoudUVtHrioNav3Md37R3EUCX6ZBTGYQz91/lov1PSi13\nrYDXxDeg+c2TPD3Qrsn7dwIF9wZ2FgXW+n3a5Qug8D6GnDjwK3F1zfuncvVNqXL7sxWPQyaa2KNC\nk72Nz7t0aYRwZxpbcbxyviCsPv+gegC/WuTwO4acDmcOP6ZyuU+bUtW5+y+02cuZu1fGZAR9XZ9/\nl4AC3T+DwT/4snHmTuG1/k/TXTx8NyGTGHI6AXNo6IkMksvy15NKuq6xnWANUG2awLfmNJ6Z9y/D\n0f0bu2Qy3tnAa0tnpEN1IQWfTCFZgBrt5S3blZHsmRTqrKSgLIDjbeB1aiS/PYUWNlpIKuJ/INcd\nGGxq6+e6tv9Y8rP3hJu5Ysdz6bd185rbl7/CuT+ANVLBB0L6z3Eb2uWURlRqlRZS1PiBW5Ra941f\nLZ1x/VKr6jlp97Eje9MZdal19vsH0tzYkXVLrcpgcDuuU9bYsNQa+6rjhNlKrdh97Ea2UmvssnHC\nShCVWnGNJmWYlFqxV35GtlIrP/1spVaO+plKrRz1M5VaOepnL7XyeYjspVY++shaauUkj6ylVib9\nm6Kmg+jdib8sAAAAAElFTkSuQmCC\n",
       "prompt_number": 70,
       "text": "\n 2  a\u22c5t                              b\u22c5t\na \u22c5\u212f    - b\u22c5(2\u22c5a + b\u22c5t\u22c5(a - b) - b)\u22c5\u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                       2                \n                (a - b)                 "
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 71,
       "text": "\n\u239b                      2                                                      \n\u239c                     p                           \u239b\u2502                 \u239b \u2148\u22c5\u03c0    \n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u239c\u2502periodic_argument\u239d\u212f   \u22c5pol\n\u239c     2                2    2          2    3     \u239d                           \n\u239d- a\u22c5b  + 2\u22c5a\u22c5b\u22c5p - a\u22c5p  + b \u22c5p - 2\u22c5b\u22c5p  + p                                  \n\n                                                                              \n             \u239e\u2502   \u03c0   \u2502                 \u239b \u2148\u22c5\u03c0                 \u239e\u2502   \u03c0\u239e   \u239b\u2502    \nar_lift(a), \u221e\u23a0\u2502 \u2264 \u2500 \u2228 \u2502periodic_argument\u239d\u212f   \u22c5polar_lift(a), \u221e\u23a0\u2502 < \u2500\u239f \u2227 \u239c\u2502peri\n                  2                                                2\u23a0   \u239d     \n                                                                              \n\n                                                                              \n             \u239b \u2148\u22c5\u03c0                 \u239e\u2502   \u03c0   \u2502                 \u239b \u2148\u22c5\u03c0           \nodic_argument\u239d\u212f   \u22c5polar_lift(b), \u221e\u23a0\u2502 < \u2500 \u2228 \u2502periodic_argument\u239d\u212f   \u22c5polar_lift\n                                        2                                     \n                                                                              \n\n             \u239e\n      \u239e\u2502   \u03c0\u239e\u239f\n(b), \u221e\u23a0\u2502 \u2264 \u2500\u239f\u239f\n           2\u23a0\u239f\n             \u23a0"
      }
     ],
     "prompt_number": 71
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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       "prompt_number": 72,
       "text": "\n \u239b     4  t\u22c5re(a)                      2    4  t\u22c5re(a)                      2 \n-\u239d- \u2148\u22c5a \u22c5\u212f       \u22c5sin(t\u22c5im(a))\u22c5\u0393(a - b)  - a \u22c5\u212f       \u22c5cos(t\u22c5im(a))\u22c5\u0393(a - b)  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n       3    t\u22c5re(a)                      2      3    t\u22c5re(a)                  \n+ 2\u22c5\u2148\u22c5a \u22c5b\u22c5\u212f       \u22c5sin(t\u22c5im(a))\u22c5\u0393(a - b)  + 2\u22c5a \u22c5b\u22c5\u212f       \u22c5cos(t\u22c5im(a))\u22c5\u0393(a \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n    2      2  2  t\u22c5re(a)                      2    2  2  t\u22c5re(a)              \n- b)  - \u2148\u22c5a \u22c5b \u22c5\u212f       \u22c5sin(t\u22c5im(a))\u22c5\u0393(a - b)  - a \u22c5b \u22c5\u212f       \u22c5cos(t\u22c5im(a))\u22c5\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n        2        2    t\u22c5re(b)                          2      2    t\u22c5re(b)    \n\u0393(a - b)  + \u2148\u22c5a\u22c5b \u22c5t\u22c5\u212f       \u22c5sin(t\u22c5im(b))\u22c5\u0393(a - b + 1)  + a\u22c5b \u22c5t\u22c5\u212f       \u22c5cos\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                    2         \n                                                             (a - b) \u22c5\u0393(a - b \n\n                      2            t\u22c5re(b)                          2         \n(t\u22c5im(b))\u22c5\u0393(a - b + 1)  + 2\u22c5\u2148\u22c5a\u22c5b\u22c5\u212f       \u22c5sin(t\u22c5im(b))\u22c5\u0393(a - b + 1)  + 2\u22c5a\u22c5b\u22c5\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    2                                                                         \n+ 1)                                                                          \n\n t\u22c5re(b)                          2      3    t\u22c5re(b)                         \n\u212f       \u22c5cos(t\u22c5im(b))\u22c5\u0393(a - b + 1)  - \u2148\u22c5b \u22c5t\u22c5\u212f       \u22c5sin(t\u22c5im(b))\u22c5\u0393(a - b + 1\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n 2    3    t\u22c5re(b)                          2      2  t\u22c5re(b)                 \n)  - b \u22c5t\u22c5\u212f       \u22c5cos(t\u22c5im(b))\u22c5\u0393(a - b + 1)  - \u2148\u22c5b \u22c5\u212f       \u22c5sin(t\u22c5im(b))\u22c5\u0393(a\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n         2    2  t\u22c5re(b)                          2\u239e             \n - b + 1)  - b \u22c5\u212f       \u22c5cos(t\u22c5im(b))\u22c5\u0393(a - b + 1) \u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                 \n                                                                 "
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "p**2/((p-a)*(p-b)**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJQAAAAvBAMAAAD3H/5RAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
       "prompt_number": 73,
       "text": "\n         2        \n        p         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                 2\n(-a + p)\u22c5(-b + p) "
      }
     ],
     "prompt_number": 73
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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       "prompt_number": 74,
       "text": "\n \u239b     4  t\u22c5re(a)                      2    4  t\u22c5re(a)                      2 \n-\u239d- \u2148\u22c5a \u22c5\u212f       \u22c5sin(t\u22c5im(a))\u22c5\u0393(a - b)  - a \u22c5\u212f       \u22c5cos(t\u22c5im(a))\u22c5\u0393(a - b)  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n       3    t\u22c5re(a)                      2      3    t\u22c5re(a)                  \n+ 2\u22c5\u2148\u22c5a \u22c5b\u22c5\u212f       \u22c5sin(t\u22c5im(a))\u22c5\u0393(a - b)  + 2\u22c5a \u22c5b\u22c5\u212f       \u22c5cos(t\u22c5im(a))\u22c5\u0393(a \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n    2      2  2  t\u22c5re(a)                      2    2  2  t\u22c5re(a)              \n- b)  - \u2148\u22c5a \u22c5b \u22c5\u212f       \u22c5sin(t\u22c5im(a))\u22c5\u0393(a - b)  - a \u22c5b \u22c5\u212f       \u22c5cos(t\u22c5im(a))\u22c5\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n        2        2    t\u22c5re(b)                          2      2    t\u22c5re(b)    \n\u0393(a - b)  + \u2148\u22c5a\u22c5b \u22c5t\u22c5\u212f       \u22c5sin(t\u22c5im(b))\u22c5\u0393(a - b + 1)  + a\u22c5b \u22c5t\u22c5\u212f       \u22c5cos\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                    2         \n                                                             (a - b) \u22c5\u0393(a - b \n\n                      2            t\u22c5re(b)                          2         \n(t\u22c5im(b))\u22c5\u0393(a - b + 1)  + 2\u22c5\u2148\u22c5a\u22c5b\u22c5\u212f       \u22c5sin(t\u22c5im(b))\u22c5\u0393(a - b + 1)  + 2\u22c5a\u22c5b\u22c5\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    2                                                                         \n+ 1)                                                                          \n\n t\u22c5re(b)                          2      3    t\u22c5re(b)                         \n\u212f       \u22c5cos(t\u22c5im(b))\u22c5\u0393(a - b + 1)  - \u2148\u22c5b \u22c5t\u22c5\u212f       \u22c5sin(t\u22c5im(b))\u22c5\u0393(a - b + 1\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n 2    3    t\u22c5re(b)                          2      2  t\u22c5re(b)                 \n)  - b \u22c5t\u22c5\u212f       \u22c5cos(t\u22c5im(b))\u22c5\u0393(a - b + 1)  - \u2148\u22c5b \u22c5\u212f       \u22c5sin(t\u22c5im(b))\u22c5\u0393(a\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n         2    2  t\u22c5re(b)                          2\u239e             \n - b + 1)  - b \u22c5\u212f       \u22c5cos(t\u22c5im(b))\u22c5\u0393(a - b + 1) \u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                 \n                                                                 "
      }
     ],
     "prompt_number": 74
    },
    {
     "cell_type": "markdown",
     "source": "19"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(a*sin(b*t)-b*sin(a*t))/(a*b*(a**2-b**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJwAAAArBAMAAAB/WfyzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nhSw5RhLn0n8A+XtcV6FPwREAAAAASUVORK5CYII=\n",
       "prompt_number": 75,
       "text": "\na\u22c5sin(b\u22c5t) - b\u22c5sin(a\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         \u239b 2    2\u239e     \n     a\u22c5b\u22c5\u239da  - b \u23a0     "
      }
     ],
     "prompt_number": 75
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 76,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 76
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/((p**2+a**2)*(p**2+b**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJAAAAApBAMAAAAorb02AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 77,
       "text": "\n         1         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239b 2    2\u239e \u239b 2    2\u239e\n\u239da  + p \u23a0\u22c5\u239db  + p \u23a0"
      }
     ],
     "prompt_number": 77
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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ZmjfohPlT2JXb6/UntUSBxanA5/BvwXY0OPMW/WkzJ5fyuLopyvqxjxNa4ZO9JSYY0oN5\nC5NaosCiUQDzaokUWCITGf5hPbcUem6KMk4M9ooYkmFuSA+mrEyqiQKLRYElbRApsEQmstQEphPB\nTAYqOTdF2TL8M2xKTkZFhjnm2bhdzRSlrkvqiQKLR4GzAEQKLJGJLNWQLQAtoZZIsXW0TP4FsHvF\nF3664ni0Ac/t+EXN5sVjTcIkUcCkQLGaF+mkKN2VSCwmk0u5c3mgrw6UGEwmtHKf5sH0YO66pJIo\nsHgUuADSIp0UnspHkNUsyORSLkMeoNdP/CcTWnlhHkgP5q5LKokCi0aB3ClTp4FIgSUykVF2LU6I\n5TF0U5RhZihOaIVj8mweTA/mrUtqiQKLRYG+Q4f2YTZ5TBQmMpFRmilOiOURdFOUYWYoTmiFYzLM\ng+nBvHVJLVFgsSog00wF6NEv+53MUDSYbXpTkl/2e1oktQ+FAqWakWZ2nLrvp0uwvO9/6R6kkPQk\nCnSlQCDNlFjNKbYCqa3EWH64qw2SyYkCH7wCgTRTghKn2DKnbjBnivrgLfmwMvgfXuZ7US1P3IQA\nAAAASUVORK5CYII=\n",
       "prompt_number": 78,
       "text": "\n   \u239b     t\u22c5im(a)                   t\u22c5im(a)                     t\u22c5im(a) + 2\u22c5t\u22c5i\n-\u2148\u22c5\u239d\u2148\u22c5a\u22c5\u212f       \u22c5sin(t\u22c5re(b)) + a\u22c5\u212f       \u22c5cos(t\u22c5re(b)) + \u2148\u22c5a\u22c5\u212f               \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\nm(b)                   t\u22c5im(a) + 2\u22c5t\u22c5im(b)                     t\u22c5im(b)        \n    \u22c5sin(t\u22c5re(b)) - a\u22c5\u212f                   \u22c5cos(t\u22c5re(b)) - \u2148\u22c5b\u22c5\u212f       \u22c5sin(t\u22c5r\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                             2\u22c5a\u22c5b\u22c5(a - b)\u22c5(a \n\n           t\u22c5im(b)                     2\u22c5t\u22c5im(a) + t\u22c5im(b)                   2\ne(a)) - b\u22c5\u212f       \u22c5cos(t\u22c5re(a)) - \u2148\u22c5b\u22c5\u212f                   \u22c5sin(t\u22c5re(a)) + b\u22c5\u212f \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n+ b)                                                                          \n\n\u22c5t\u22c5im(a) + t\u22c5im(b)             \u239e  -t\u22c5im(a) - t\u22c5im(b)             \n                  \u22c5cos(t\u22c5re(a))\u23a0\u22c5\u212f                  \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                 "
      }
     ],
     "prompt_number": 78
    },
    {
     "cell_type": "markdown",
     "source": "20"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(cos(b*t)-cos(a*t))/(a**2-b**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJkAAAAnBAMAAADusveMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
       "prompt_number": 79,
       "text": "\n-cos(a\u22c5t) + cos(b\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n       2    2       \n      a  - b        "
      }
     ],
     "prompt_number": 79
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 80,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 80
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "p/((p**2+a**2)*(p**2+b**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJAAAAAlBAMAAABfb31NAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
       "prompt_number": 81,
       "text": "\n         p         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239b 2    2\u239e \u239b 2    2\u239e\n\u239da  + p \u23a0\u22c5\u239db  + p \u23a0"
      }
     ],
     "prompt_number": 81
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABWgAAAAuBAMAAABHZO2hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 82,
       "text": "\n\u239b   t\u22c5im(a)                 t\u22c5im(a)                   t\u22c5im(b)                 \n\u239d\u2148\u22c5\u212f       \u22c5sin(t\u22c5re(b)) + \u212f       \u22c5cos(t\u22c5re(b)) - \u2148\u22c5\u212f       \u22c5sin(t\u22c5re(a)) - \u212f\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\nt\u22c5im(b)                   t\u22c5im(a) + 2\u22c5t\u22c5im(b)                 t\u22c5im(a) + 2\u22c5t\u22c5im\n       \u22c5cos(t\u22c5re(a)) - \u2148\u22c5\u212f                   \u22c5sin(t\u22c5re(b)) + \u212f                \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                     2\u22c5(a - b)\u22c5(a + b)        \n\n(b)                   2\u22c5t\u22c5im(a) + t\u22c5im(b)                 2\u22c5t\u22c5im(a) + t\u22c5im(b) \n   \u22c5cos(t\u22c5re(b)) + \u2148\u22c5\u212f                   \u22c5sin(t\u22c5re(a)) - \u212f                   \u22c5\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n            \u239e  -t\u22c5im(a) - t\u22c5im(b)             \ncos(t\u22c5re(a))\u23a0\u22c5\u212f                  \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                              "
      }
     ],
     "prompt_number": 82
    },
    {
     "cell_type": "markdown",
     "source": "21"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "cos(a*t)**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAUBAMAAAAgmk0yAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVJl2Zs0i7zIQRN2r\nibuk4LwrAAABY0lEQVQoFW1Rv0vDQBT+StI0aVNb1M0OpXUREYLpEqdURehkV3HJoAg6ZHW8QXEz\nRQeXQoP+AcbVKeCikwU3QRD8B5xEF+u7XJK2kg/ufT/uXfJyASagmo8TLkte4qyXlY8zDxWfO5uX\nCHbMCakYOKR1XgQuEpHyClft2CoMOSPdEkKucw6EwcwQ2Il1QjegSdNzfKhWsiVYtfYYjeELB4/4\nlNasaWNuw0Z7vuuORhRUaEmtBVzvNiMjdZXjch1vxSGeaIswoPWBZV6AQg+DqrbkOnAP7+FTQjgA\n5AAN4JtMycEtA7bpafbzLymOIzo5xB3UTzIllnZYSu0rasAV4PraD+R38tFboLgM7gnDi+igSb2e\nHFR1IycmDbBVDtGQQ6yJjjxDwdkPjXy1A5xTVjN9rK/asklfHEEyUFzsPDi5JgM2RTZdteTWefz/\n1kXr6/gE3VIWdPo5MfqJmGbNSL0F/AEhukYATWbTCAAAAABJRU5ErkJggg==\n",
       "prompt_number": 83,
       "text": "\n   2     \ncos (a\u22c5t)"
      }
     ],
     "prompt_number": 83
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#laplace_transform(_, t, p)\n# Python exception",
     "language": "python",
     "outputs": [],
     "prompt_number": 84
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(p**2+2*a**2)/(p*(p**2+4*a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAF4AAAAsBAMAAAAeFhKXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 85,
       "text": "\n     2    2  \n  2\u22c5a  + p   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  \u239b   2    2\u239e\np\u22c5\u239d4\u22c5a  + p \u23a0"
      }
     ],
     "prompt_number": 85
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAv8AAAAmBAMAAAChe+nfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 86,
       "text": "\n \u239b   4\u22c5t\u22c5im(a)                   4\u22c5t\u22c5im(a)                     2\u22c5t\u22c5im(a)      \n-\u239d\u2148\u22c5\u212f         \u22c5sin(2\u22c5t\u22c5re(a)) - \u212f         \u22c5cos(2\u22c5t\u22c5re(a)) - 2\u22c5\u212f          - \u2148\u22c5s\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                  4           \n\n                              \u239e  -2\u22c5t\u22c5im(a)             \nin(2\u22c5t\u22c5re(a)) - cos(2\u22c5t\u22c5re(a))\u23a0\u22c5\u212f          \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                        "
      }
     ],
     "prompt_number": 86
    },
    {
     "cell_type": "markdown",
     "source": "22"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sin(a*t)**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD8AAAAUBAMAAADIGvgZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZlTNRImrEN27MiJ2\n75lZwTvPAAABVklEQVQoFW1QMUvDQBT+ksZEY0qtggVRKLqImyDoGBDBQaSDP8BZhGRyUhP8BY4i\nDgeC4CBNdXGRJFMXf4QOgkMQRUGoQn13ueZa6YN77/u+993d4wEq7Mc9RYahQ9z4w/RCa6LicjJR\nKAoJyUYQETB5yuNY1jlfghdeJyUZZ9CTHC9Ig/bEecgTxVgDWM1hL2+BnL1bYqC01xLVbrcYjeBK\nsUn1EtDmp+wztv5TzRB3u6RV6DjpIrbXMkE2YPoBs34Ry0ECMlzglSeg7GPTNVBh+OBJxAO9GuIN\n6BAdjWB2z3mvI5cE3NO9Bk5gv3MDg3/rRQOGIyB2rW9op2SgL2YwkgwYaMimr4V1M9HFkNNwXDJ8\nFV+UGMrR3W5SqteAa2A2TW1v5epzyVumJymcBMZzbT/SMwZUhTSYrLCP/1t13tlRBqOhsEKm3Bgp\nB0rtQ1ZSkDbwBzeCUIGBlL/7AAAAAElFTkSuQmCC\n",
       "prompt_number": 87,
       "text": "\n   2     \nsin (a\u22c5t)"
      }
     ],
     "prompt_number": 87
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#laplace_transform(_, t, p)\n# Python exception",
     "language": "python",
     "outputs": [],
     "prompt_number": 88
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2*a**2/(p*(p**2+4*a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAsBAMAAAB7++DoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nKwT+AXtGq1RoBBGFAAAAAElFTkSuQmCC\n",
       "prompt_number": 89,
       "text": "\n        2    \n     2\u22c5a     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  \u239b   2    2\u239e\np\u22c5\u239d4\u22c5a  + p \u23a0"
      }
     ],
     "prompt_number": 89
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAvAAAAAmBAMAAABQcLJSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzTcgqeirl8wGwN6AvMHAdcnA/wFfUWErw+gg9wAAAABJRU5ErkJggg==\n",
       "prompt_number": 90,
       "text": "\n\u239b   4\u22c5t\u22c5im(a)                   4\u22c5t\u22c5im(a)                     2\u22c5t\u22c5im(a)       \n\u239d\u2148\u22c5\u212f         \u22c5sin(2\u22c5t\u22c5re(a)) - \u212f         \u22c5cos(2\u22c5t\u22c5re(a)) + 2\u22c5\u212f          - \u2148\u22c5si\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                  4           \n\n                             \u239e  -2\u22c5t\u22c5im(a)             \nn(2\u22c5t\u22c5re(a)) - cos(2\u22c5t\u22c5re(a))\u23a0\u22c5\u212f          \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                       "
      }
     ],
     "prompt_number": 90
    },
    {
     "cell_type": "markdown",
     "source": "23"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "cosh(a*t)**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEsAAAAUBAMAAAA3uN37AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+AOBVFkh6EjbWgAAAABJRU5ErkJggg==\n",
       "prompt_number": 91,
       "text": "\n    2     \ncosh (a\u22c5t)"
      }
     ],
     "prompt_number": 91
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 92,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 92
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(p**2-2*a**2)/(p*(p**2-4*a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAsBAMAAAByEECSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 93,
       "text": "\n       2    2  \n  - 2\u22c5a  + p   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  \u239b     2    2\u239e\np\u22c5\u239d- 4\u22c5a  + p \u23a0"
      }
     ],
     "prompt_number": 93
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAvAAAAAmBAMAAABQcLJSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 94,
       "text": "\n\u239b   4\u22c5t\u22c5re(a)                   4\u22c5t\u22c5re(a)                     2\u22c5t\u22c5re(a)       \n\u239d\u2148\u22c5\u212f         \u22c5sin(2\u22c5t\u22c5im(a)) + \u212f         \u22c5cos(2\u22c5t\u22c5im(a)) + 2\u22c5\u212f          - \u2148\u22c5si\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                  4           \n\n                             \u239e  -2\u22c5t\u22c5re(a)             \nn(2\u22c5t\u22c5im(a)) + cos(2\u22c5t\u22c5im(a))\u23a0\u22c5\u212f          \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                       "
      }
     ],
     "prompt_number": 94
    },
    {
     "cell_type": "markdown",
     "source": "24"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sinh(a*t)**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAAUBAMAAAAzTQ3GAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 95,
       "text": "\n    2     \nsinh (a\u22c5t)"
      }
     ],
     "prompt_number": 95
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 96,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 96
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2*a**2/(p*(p**2-4*a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHYAAAAsBAMAAABCnFGzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 97,
       "text": "\n         2     \n      2\u22c5a      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  \u239b     2    2\u239e\np\u22c5\u239d- 4\u22c5a  + p \u23a0"
      }
     ],
     "prompt_number": 97
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAvAAAAAmBAMAAABQcLJSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73ZnN\nZu+9Cd19AAAJkUlEQVRoBe1Zf2wbVx3/XO5sX+JccipTN1EVe+tUDSFRM7VDWlt6qmBTBSOGrRNj\nVXNpmbVSgrwKCtqYsDQYP7TWGWIrYVtl1GyA0KQAWsYWsmUlaDAJ4ZaSlGZeU2mMSWias1idy1KF\n73vv7nqXvpyTxlarKO+Pu+99f3y+3+/H7957tgFlDZbHZWBAu/P9y5B1OSUxsEz8ZZoGHvGRy1TA\nlZ9WsRtRo0f8oUagLw3MTCPacImPpxuBvjQwm7MN6MMlflMDsJcKpHKmAZ24xB+5GPsTuC13sXbB\nmjrB8LyqqZ1ccAGLDzguhVhcYw7x0eEA9h1QrOiKDduC2oDLfB7qBMNS3bLLwlXoWnPHk/NJXGef\ntlwAUMtcv3h+HOLbUn5o9U3Ei3F1QnffB79x/nKdYFhCxdbOYAj9n0K+EQtujZ4i/QGHlfi8uWh+\nHGq3BJBXj6Alp54sY2dAvdCHOsGwtFoZU/gVNucRmVhoGYv318sBjA60pxbJj37X/37PMT8aQLZH\n1J278YVJHEdmTy5gYg/FizRyZQ0YGDIcqRKwlOqGt1KRhGkko5mfyQKDOimMVOnFSa1C+Y7nxAQN\niYfC+Ak4Bx44WiDPM357LDeCfdC7zr5kxu2E38LlppynWj3a5cqrXMG714ABbvBclVLGdB6U4Gvt\nucTTsSQeTuw9gwElOAE9F7+wIGwRGBLyOz80yc+F8TPLF7jxUVfFU/g7VKquid2vxgitp8Z9kx3I\nm59zLG+4zGAM+OcLWeB70NP4Ogl8xGxH8G5SGM8KGFkYdx8CdAvfR/Q/rmW/K/A7z8Skcdp2MJa4\nqqxPasPcdPHF8M5988IOAoSFnHJdxwcHB4sw+oP8fPbdx7Wd59KuE7+LWvhVddmlFMQafB3Gyv6g\n0cx/U30K9HIH7sc9juU7putSQDSnnQceQxyIVFz1K0K41X2GFMazAi3AvegoojmNfqDTckytXiJS\niEwkqLbSYpnIa+VYIZp2XGffYk4NtbC17OxIeg4rJ7gF4iaYQX46TUQmg5iiFn71pjWlINbg6zA6\nEYzqQ3l9RB3O40YcKQZNoKndZuPjoN2O/b7z4tWOQ7fwe9DnHgbDP/i30DRBOxWIyfgvnMioP6HI\nRJC3XHOiXbXSm/SymmzZ7MshF2lShWDH/ClcgLCQhOV6sbvWdcAK8iMh3hfgEc+m+hTNpgvpIwWf\nH7DyvdRQ1vhMqqUUL+3NBkxAUwotaXSadNBgxGvfduzOmuQjPhQGoL8CnmYTpQOMeNyaE0CBhYRn\nIr0+M1OOj+HE7Xi2ODT6vPAMuYZjS4kPC2kPkJCfmUGQn3kSTymINfg6bBr2mjC6/+DIn4aSEqLx\nrde1n1ubqqUBem7nusPqH39jq0Vg1SiQGSu4ejjEy2DwQKaI/aO0VD84TCBUAtA8sfG1AdAUVoZs\nRLvvd/XMJsZhYHQ8J2TNwocc9UU3At3Wp/79wLhNpnBsh/jYQDdi3d2g9mqEtFOjvjLYA3CBH5qG\nfKnJjOeU0l7z1HvofHRbH2uTrurQWJU6Y6yWwVhzqmMQaO7hN3Z5DA97sitsQJOZsJSztNcC/JhD\ncrNNC4CNXr3YbMcLQBvZaDjEy2Ciw/qbkX5sV9O4lpguM/eEjcdpDmSohAMkvUiqM0zvDZYprdve\n81wCA1UqhKfTYlsD2yH+azho3Y1ImrUXHkIrHpE0dxmdz23fMYl4v34m3mMMUxlxKBWnotdhVDGI\ngyIFY406nGGDpNYkXfig7ydfcmXvfnNKRbuF8+yCg0zd3M9lkv4KnKKUtDnlmMEhXgqTyCq/zeeQ\nf+gIUrR6TDD3JwCxPbf2wChQgaDvSf5BmXYikvOrpLLKQCuguUl4NbAF8cr7WHVNFXqVtRceQise\nwsoQM77DwrNqkT74J7Ce2hIVURa9qlXQUuQpGIO+DpuTtHZfR2NN+1PdOTIFR9PMTxnP1BYR9mNm\nuwdsaWbjuyn8Gy1ZIt6iHbK397Xe3iStOzKYLeRxmmzFdedI0gugfSbNVz2SYlNoS+OXJPWRrZcN\nmx5YpnOjJ5k0a/yIudDbIuoGGChVaLHtKwwbL/Q++even9BBiT4howycZ+2Fh7TZlFxeBq9KEH+6\nVHoeqzNVbDHpg6qIiugsqVeN86XxLE/BWaMOndHU40r5lCv57uYP1uWoI0H8R8gQsxX8i8q+j7ie\n1KeQMGctNVIYl/gufV+VloMJAtoP06DcYxT/splPKexMFlhqKBPhz2MoDNQlvga2mPEu8WdZe+Hl\nsDU+rAyHeCrT6KMzTMsx8q+IigTxZdYBq4pY83d4gfgElcBsgXECEdsjvp1MNyGKtcjitI1YRZvE\nqdiszVUKk6APPW8hf5uFOwmEilFt2E12DMZZenWsDtMoZP1bj8jETmMmieHDYKAu8TWwvaUGP6xA\nr7D2wsuhyhFWhiCeZpvWntWr2egzVG9FVKRM0YxnH5rFUzDWxNbP24kU+I0u9G/LP1zZux9HNEXE\nT/GlptWC9lRpD141TOhH6ViCt/WjUWCbcBebqxQmWsDJSBKDRhK7yXmEYkqjPa3ZY8AjSRxGW+6R\npA2tx8tLuy7LdAhGyqeTixyUv5PsBQnHFsTjqzByX6T9ibUXHsJf4JAyBPGRHpxMpGJVW/mASqg4\nFe1BtIK7KBVPwVjzdRgru90oQ6PkMWuMdndr6/60dfob6/5GyzJ9gZqZOYcHvkJekT8fzWFjacW9\nQEkECeKlMNiXSWF8b9HI0JGSDqHAuzMzPbEBix527LiBDl7H1uZgMJs7eKZVJYKvNRjo1un1617d\nOk0A4dgO8bG1u2EMDIHaI/SwkA6THOYug/1k8BL9ZDCesWJrT+zK4npg6/RmXlEqNjQ2nTUGGFuU\ngrHGO9STpKCXgV3nN5SC3O8Vof6k3CrRfliiI1Ura3KxIxxbzUrww0LekPhfgspJwTukUxwb3k9U\n84C7XeqjpqXqEGUkJTVukGoXqLwE7LCQ6xaYfg53JwXvcJcg/mNzuMrU9G1DMlZIdDVU18rsSlKm\nXbDuErBDQt5ZcH55AE+hJMmobRTE3yx3lGoVW6bukinDdStl5qj0Y5V5huouAXvuEPZluC6Dp+Ad\nRuOC+DarLsBLFCQqSKpnd7ZDfDRZT9SlhtWWqndHquUQj7frDb2U8PbUvZnVcInnX2Xrjr9EAOs/\nK7+5/ctPi9eoKbtESGpAG03FRoD2OKB/aQD4EoHc1Yg+vL9ALuEc3oh6rkBMvb8BRWkvf2A1AHYZ\ncpmBK5KB/wO5l1pspKcxRgAAAABJRU5ErkJggg==\n",
       "prompt_number": 98,
       "text": "\n\u239b   4\u22c5t\u22c5re(a)                   4\u22c5t\u22c5re(a)                     2\u22c5t\u22c5re(a)       \n\u239d\u2148\u22c5\u212f         \u22c5sin(2\u22c5t\u22c5im(a)) + \u212f         \u22c5cos(2\u22c5t\u22c5im(a)) - 2\u22c5\u212f          - \u2148\u22c5si\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                  4           \n\n                             \u239e  -2\u22c5t\u22c5re(a)             \nn(2\u22c5t\u22c5im(a)) + cos(2\u22c5t\u22c5im(a))\u23a0\u22c5\u212f          \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                       "
      }
     ],
     "prompt_number": 98
    },
    {
     "cell_type": "markdown",
     "source": "25"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sin(a*t)*sinh(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAASBAMAAACEDMyZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nPfwFhVubkIMnfh8AAAAASUVORK5CYII=\n",
       "prompt_number": 99,
       "text": "sin(a\u22c5t)\u22c5sinh(a\u22c5t)"
      }
     ],
     "prompt_number": 99
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#laplace_transform(_, t, p)\n# Python exception",
     "language": "python",
     "outputs": [],
     "prompt_number": 100
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2*a**2*p/(p**2+4*a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE0AAAAsBAMAAADSjyh8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVx2hqqb9F6Spk3I+bKmrAAAAAElFTkSuQmCC\n",
       "prompt_number": 101,
       "text": "\n     2   \n  2\u22c5a \u22c5p \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   2    2\n4\u22c5a  + p "
      }
     ],
     "prompt_number": 101
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAr0AAAAhBAMAAAAmMQ76AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 102,
       "text": "\n  2 \u239b   4\u22c5t\u22c5im(a)                   4\u22c5t\u22c5im(a)                                 \n-a \u22c5\u239d\u2148\u22c5\u212f         \u22c5sin(2\u22c5t\u22c5re(a)) - \u212f         \u22c5cos(2\u22c5t\u22c5re(a)) - \u2148\u22c5sin(2\u22c5t\u22c5re(a)\n\n                  \u239e  -2\u22c5t\u22c5im(a)             \n) - cos(2\u22c5t\u22c5re(a))\u23a0\u22c5\u212f          \u22c5Heaviside(t)"
      }
     ],
     "prompt_number": 102
    },
    {
     "cell_type": "markdown",
     "source": "26"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sin(a*t)*cosh(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAASBAMAAACA+RykAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnRgOpwG4AAAAAElFTkSuQmCC\n",
       "prompt_number": 103,
       "text": "sin(a\u22c5t)\u22c5cosh(a\u22c5t)"
      }
     ],
     "prompt_number": 103
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#laplace_transform(_, t, p)\n# Python exception",
     "language": "python",
     "outputs": [],
     "prompt_number": 104
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a*(p**2+2*a**2)/(p**4+4*a**4)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAuBAMAAAAl5AEXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 105,
       "text": "\n  \u239b   2    2\u239e\na\u22c5\u239d2\u22c5a  + p \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     4    4  \n  4\u22c5a  + p   "
      }
     ],
     "prompt_number": 105
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAB2sAAAAuBAMAAAD0P+9jAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDAAAAABJRU5ErkJggg==\n",
       "prompt_number": 106,
       "text": "\n  \u239b   -t\u22c5re(a) - t\u22c5im(a)                           -t\u22c5re(a) - t\u22c5im(a)         \n\u2148\u22c5\u239d\u2148\u22c5\u212f                  \u22c5sin(t\u22c5re(a) - t\u22c5im(a)) + \u212f                  \u22c5cos(t\u22c5re\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n                    -t\u22c5re(a) + t\u22c5im(a)                           -t\u22c5re(a) + t\u22c5\n(a) - t\u22c5im(a)) + \u2148\u22c5\u212f                  \u22c5sin(t\u22c5re(a) + t\u22c5im(a)) - \u212f             \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\nim(a)                             t\u22c5re(a) - t\u22c5im(a)                           \n     \u22c5cos(t\u22c5re(a) + t\u22c5im(a)) + \u2148\u22c5\u212f                 \u22c5sin(t\u22c5re(a) + t\u22c5im(a)) + \u212f\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                        2\u22c5(-1 + \u2148)\u22c5(1 + \u2148)                                    \n\nt\u22c5re(a) - t\u22c5im(a)                             t\u22c5re(a) + t\u22c5im(a)               \n                 \u22c5cos(t\u22c5re(a) + t\u22c5im(a)) + \u2148\u22c5\u212f                 \u22c5sin(t\u22c5re(a) - \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n            t\u22c5re(a) + t\u22c5im(a)                       \u239e             \nt\u22c5im(a)) - \u212f                 \u22c5cos(t\u22c5re(a) - t\u22c5im(a))\u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                  "
      }
     ],
     "prompt_number": 106
    },
    {
     "cell_type": "markdown",
     "source": "27"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "cos(a*t)*sinh(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAASBAMAAACA+RykAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
       "prompt_number": 107,
       "text": "cos(a\u22c5t)\u22c5sinh(a\u22c5t)"
      }
     ],
     "prompt_number": 107
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#laplace_transform(_, t, p)\n# Python exception",
     "language": "python",
     "outputs": [],
     "prompt_number": 108
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a*(p**2-2*a**2)/(p**4+4*a**4)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAAuBAMAAAA7LTGkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n39hpYdDF4QAAAABJRU5ErkJggg==\n",
       "prompt_number": 109,
       "text": "\n  \u239b     2    2\u239e\na\u22c5\u239d- 2\u22c5a  + p \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n      4    4   \n   4\u22c5a  + p    "
      }
     ],
     "prompt_number": 109
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAB3oAAAAxBAMAAADO07X7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nd1l5vXTlpFbg/10sgV0SNZGGAAAAAElFTkSuQmCC\n",
       "prompt_number": 110,
       "text": "\n   \u239b -t\u22c5re(a) - t\u22c5im(a)                             -t\u22c5re(a) - t\u22c5im(a)        \n-\u2148\u22c5\u239d\u212f                  \u22c5sin(t\u22c5re(a) - t\u22c5im(a)) - \u2148\u22c5\u212f                  \u22c5cos(t\u22c5r\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                   -t\u22c5re(a) + t\u22c5im(a)                             -t\u22c5re(a) + t\ne(a) - t\u22c5im(a)) - \u212f                  \u22c5sin(t\u22c5re(a) + t\u22c5im(a)) - \u2148\u22c5\u212f            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n\u22c5im(a)                           t\u22c5re(a) - t\u22c5im(a)                            \n      \u22c5cos(t\u22c5re(a) + t\u22c5im(a)) - \u212f                 \u22c5sin(t\u22c5re(a) + t\u22c5im(a)) + \u2148\u22c5\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                 2        2                                   \n                         (-1 + \u2148) \u22c5(1 + \u2148)                                    \n\n t\u22c5re(a) - t\u22c5im(a)                           t\u22c5re(a) + t\u22c5im(a)                \n\u212f                 \u22c5cos(t\u22c5re(a) + t\u22c5im(a)) + \u212f                 \u22c5sin(t\u22c5re(a) - t\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n             t\u22c5re(a) + t\u22c5im(a)                       \u239e             \n\u22c5im(a)) + \u2148\u22c5\u212f                 \u22c5cos(t\u22c5re(a) - t\u22c5im(a))\u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                   \n                                                                   "
      }
     ],
     "prompt_number": 110
    },
    {
     "cell_type": "markdown",
     "source": "28"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "cos(a*t)*cosh(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIEAAAASBAMAAACOfKCOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 111,
       "text": "cos(a\u22c5t)\u22c5cosh(a\u22c5t)"
      }
     ],
     "prompt_number": 111
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#laplace_transform(_, t, p)\n# Python exception",
     "language": "python",
     "outputs": [],
     "prompt_number": 112
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "p**3/(p**4+4*a**4)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAsBAMAAAAjhHPxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 113,
       "text": "\n     3   \n    p    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   4    4\n4\u22c5a  + p "
      }
     ],
     "prompt_number": 113
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "#inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [],
     "prompt_number": 114
    },
    {
     "cell_type": "markdown",
     "source": "29"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "t/2*sin(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAlBAMAAAAaQhKKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nbEWJ8llMhZuBVwHTDsprCthgp9P6HfsE8t9vhKf3YcwAAAAASUVORK5CYII=\n",
       "prompt_number": 115,
       "text": "\nt\u22c5sin(a\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    2     "
      }
     ],
     "prompt_number": 115
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 116,
       "text": "\n\u239b     a           \u2502                 \u239b          2      \u239e\u2502    \u239e\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 = 0\u239f\n\u239c           2                                               \u239f\n\u239c   \u239b 2    \u239e                                                \u239f\n\u239c 3 \u239ca     \u239f                                                \u239f\n\u239cp \u22c5\u239c\u2500\u2500 + 1\u239f                                                \u239f\n\u239c   \u239c 2    \u239f                                                \u239f\n\u239d   \u239dp     \u23a0                                                \u23a0"
      }
     ],
     "prompt_number": 116
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "ratsimp(_[0])",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAAlBAMAAACQbYvLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 117,
       "text": "\n       a\u22c5p       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 4      2  2    4\na  + 2\u22c5a \u22c5p  + p "
      }
     ],
     "prompt_number": 117
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#inverse_laplace_transform(_, p, t)\n# ValueError: too many values to unpack",
     "language": "python",
     "outputs": [],
     "prompt_number": 118
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a*p/(p**2+a**2)**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAnBAMAAAC8vQiCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdmZU3buZ74kiRBAy\nzauYRQn6AAAB9klEQVQ4Eb1Rz2sTQRh9u2mS7s5oN4KCCBJiLUI9pOAfsCAeBA+xkKMSRIuQS4/+\nuPTg0UPoXRIvtiDCtngRxGxPHuMfINoe/XGIF6GNsH4z883S3YQ0pz6Yne+9782bmR0gB+fcDRQu\n7C5FOX2M3sVD7PaDUmuskxXcEGt4O8B8mNXHWHkDN4HzKDbGWlmh2xMjYBW0YDoGkRvWcYitky5T\n7rzaD+Q/tf90+A+WDzru6rV4uo27Z4OZbEC5N6Oxe9JNOMf/3p4x8ZRsyWwYntJx8tvsXY7z0kQu\nAvl8YiMvyiGOlOaaxoR8Wfmtu7E4VJ4f2jgpfwefI9GidmmDPm6dPoDN14Q/Ayz0sEJkSQlzrHI+\npFmoVIl+B2ci+IEgplZo6HygUGWupseAU8Xeh+tU/6ShofOzRrdFsV+8JFEvqEblSsj5qXGlXWvj\nNSJjII8gYzEohZxvjWL5D9bkpc0Y4J/trQOLKLU43xr9j0O86CYJRd2mQfBC4Anm6poAj5r3bjXv\nE3EaaBrtjpnEOrwj9OksDL51sUrrNew7DyFHWCxYn/09Cx2fHXQJja/AU++bkzcOYj6ObHCrAryp\nfbqYN16tvTSSW+XW+9RiCt/c68DK9IQG8z0rZeZnlm3bAr/S6ljhjZiI/VTdSatjxbu/vKPD838t\nb5LyLM1PzQAAAABJRU5ErkJggg==\n",
       "prompt_number": 119,
       "text": "\n   a\u22c5p    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         2\n\u239b 2    2\u239e \n\u239da  + p \u23a0 "
      }
     ],
     "prompt_number": 119
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#inverse_laplace_transform(_, p, t)\n# Same error here",
     "language": "python",
     "outputs": [],
     "prompt_number": 120
    },
    {
     "cell_type": "markdown",
     "source": "30"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "t/2*sinh(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFcAAAAlBAMAAADFDGiVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 121,
       "text": "\nt\u22c5sinh(a\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     2     "
      }
     ],
     "prompt_number": 121
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 122,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 122
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a*p/(p**2-a**2)**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAF0AAAAnBAMAAACf5llXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\noFi8pPEHdaMNxaL9QsNr6kYbisWQOs4vrPuZzRn9HiIAAAAASUVORK5CYII=\n",
       "prompt_number": 123,
       "text": "\n    a\u22c5p     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n           2\n\u239b   2    2\u239e \n\u239d- a  + p \u23a0 "
      }
     ],
     "prompt_number": 123
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAoUAAAAmBAMAAABAEPyOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRI25ireqQHcK/AdiWMXgRJw+IwAAAABJRU5ErkJggg==\n",
       "prompt_number": 124,
       "text": "\n  \u239b   2\u22c5t\u22c5re(a)                 2\u22c5t\u22c5re(a)                                     \nt\u22c5\u239d\u2148\u22c5\u212f         \u22c5sin(t\u22c5im(a)) + \u212f         \u22c5cos(t\u22c5im(a)) + \u2148\u22c5sin(t\u22c5im(a)) - cos(\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                      4                       \n\n        \u239e  -t\u22c5re(a)             \nt\u22c5im(a))\u23a0\u22c5\u212f        \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                "
      }
     ],
     "prompt_number": 124
    },
    {
     "cell_type": "markdown",
     "source": "31"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(b*sinh(a*t)-a*sinh(b*t))/(a**2-b**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAL4AAAAnBAMAAABDM+8lAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 125,
       "text": "\n-a\u22c5sinh(b\u22c5t) + b\u22c5sinh(a\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          2    2          \n         a  - b           "
      }
     ],
     "prompt_number": 125
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 126,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 126
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a*b/((p**2-a**2)*(p**2-b**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKwAAAAqBAMAAADc9q0uAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nBIKMheX+AAAAAElFTkSuQmCC\n",
       "prompt_number": 127,
       "text": "\n          a\u22c5b          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239b   2    2\u239e \u239b   2    2\u239e\n\u239d- a  + p \u23a0\u22c5\u239d- b  + p \u23a0"
      }
     ],
     "prompt_number": 127
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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kD1EV6KyaKdBzBfob8EUEtTyLi1Nop//u4A9xsfJWflfO2RId2S1TYBkqUBpnp+RtjFLr\nb6FtHX4V23iRReT349guzEpjds8UWL4KnIrUcK/NT2kKaQZnfQ2gjZ2iRX08v9VztsIZWS1TYPkp\ngCdViRO6aupGmx0oxY4taCJndooWcef5rZ6ztfw8yhhlCoQK9OE+O7cLT+mg87eUEpz1hSdsuXRQ\nGCs8v4vKOVvKlKyaKbDsFKBDf+kcKH5KE4g/8uEnU7GzqvCIpX5+MlR4qpY8Z2vZeZMRyhTQFCg3\nHHZekzilKejDQ2Z28rO+8L9dxclQ2Mmf3+o5W8GMrJIpsPwUuBfySKoL4pSmgGB41he+X4qTobBT\n5DfgDoW9lAbjs0qmwDJUoHDa5JlIC/8znJ/SFFIMzvpagUdK00FhrIvnt3rOVjgjq2UKLDcF+ufn\nDyAnPK+JnywV8gvO+sIjlvjJUNTH81s9ZyuckdUyBZapApbzmhhT+v28PGKJGZRTtbLfzy/TYGa0\nIgpU2hETGfixSpETtvjYd+7fya3cMmOmQHoFrrAO5ccq8SOWzAE9OijKhM3amQJvgwJXWTH5sUoO\nf7U0RuR9w5A1l6bA/wAYA4g11Ba1gwAAAABJRU5ErkJggg==\n",
       "prompt_number": 128,
       "text": "\n \u239b     t\u22c5re(a)                   t\u22c5re(a)                     t\u22c5re(a) + 2\u22c5t\u22c5re(\n-\u239d\u2148\u22c5a\u22c5\u212f       \u22c5sin(t\u22c5im(b)) - a\u22c5\u212f       \u22c5cos(t\u22c5im(b)) + \u2148\u22c5a\u22c5\u212f                 \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\nb)                   t\u22c5re(a) + 2\u22c5t\u22c5re(b)                     t\u22c5re(b)          \n  \u22c5sin(t\u22c5im(b)) + a\u22c5\u212f                   \u22c5cos(t\u22c5im(b)) - \u2148\u22c5b\u22c5\u212f       \u22c5sin(t\u22c5im(\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                              2\u22c5(a - b)\u22c5(a + b\n\n         t\u22c5re(b)                     2\u22c5t\u22c5re(a) + t\u22c5re(b)                   2\u22c5t\na)) + b\u22c5\u212f       \u22c5cos(t\u22c5im(a)) - \u2148\u22c5b\u22c5\u212f                   \u22c5sin(t\u22c5im(a)) - b\u22c5\u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n)                                                                             \n\n\u22c5re(a) + t\u22c5re(b)             \u239e  -t\u22c5re(a) - t\u22c5re(b)             \n                \u22c5cos(t\u22c5im(a))\u23a0\u22c5\u212f                  \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                               "
      }
     ],
     "prompt_number": 128
    },
    {
     "cell_type": "markdown",
     "source": "32"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(cosh(a*t)-cosh(b*t))/(a**2-b**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ0AAAAnBAMAAADnWVf2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nLZx7dvwDs8EnNkg4o5QAAAAASUVORK5CYII=\n",
       "prompt_number": 129,
       "text": "\ncosh(a\u22c5t) - cosh(b\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        2    2       \n       a  - b        "
      }
     ],
     "prompt_number": 129
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 130,
       "text": "(nan, -\u221e, True)"
      }
     ],
     "prompt_number": 130
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "p/((p**2-a**2)*(p**2-b**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKwAAAAlBAMAAAAtoB/7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9int1dBRi2TFYgv0NyMw+hoV2pmXAAAAAElFTkSuQmCC\n",
       "prompt_number": 131,
       "text": "\n           p           \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239b   2    2\u239e \u239b   2    2\u239e\n\u239d- a  + p \u23a0\u22c5\u239d- b  + p \u23a0"
      }
     ],
     "prompt_number": 131
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABWgAAAAuBAMAAABHZO2hAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\niMGsctiYK3AGTRjs/0vbfwGGeQTL1ar4IQAAAABJRU5ErkJggg==\n",
       "prompt_number": 132,
       "text": "\n\u239b   t\u22c5re(a)                 t\u22c5re(a)                   t\u22c5re(b)                 \n\u239d\u2148\u22c5\u212f       \u22c5sin(t\u22c5im(b)) - \u212f       \u22c5cos(t\u22c5im(b)) - \u2148\u22c5\u212f       \u22c5sin(t\u22c5im(a)) + \u212f\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\nt\u22c5re(b)                   t\u22c5re(a) + 2\u22c5t\u22c5re(b)                 t\u22c5re(a) + 2\u22c5t\u22c5re\n       \u22c5cos(t\u22c5im(a)) - \u2148\u22c5\u212f                   \u22c5sin(t\u22c5im(b)) - \u212f                \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                     2\u22c5(a - b)\u22c5(a + b)        \n\n(b)                   2\u22c5t\u22c5re(a) + t\u22c5re(b)                 2\u22c5t\u22c5re(a) + t\u22c5re(b) \n   \u22c5cos(t\u22c5im(b)) + \u2148\u22c5\u212f                   \u22c5sin(t\u22c5im(a)) + \u212f                   \u22c5\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n\n            \u239e  -t\u22c5re(a) - t\u22c5re(b)             \ncos(t\u22c5im(a))\u23a0\u22c5\u212f                  \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                              "
      }
     ],
     "prompt_number": 132
    },
    {
     "cell_type": "markdown",
     "source": "33"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/sqrt(pi*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAArBAMAAADMPRbsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73ZnN\n72bDLMX7AAABCUlEQVQ4Ed2Tv0vDUBSFvyaxmEhtUBy6dRBxrP9BEJw7uDh2cLe7g2nRVXRwdpYu\nDuLk0M1/o6uboKBO8dTk/ajQsVB6hve+c+6Fe3nwwGmr49ino7O+bz1urnjh6n405028R1haLOZo\n0Qt3NdfMmHJRvJf21qS6fY4eXcFn4pxLUxI7HcKdcWKnffgwTmy13iaqtkDc0lLff8UgC19GvbIv\nyNjd2T4ozRDiKmdIlF7XcwTwBM1Md8WcBBDqi9RuoKvcMONELh6zpu43oeGw3cjV9klL2R59y/Ek\nmSh64Fjn62ZqOUk3ejIXA43g/FRHxVOSGs+dEv4x9R+bzzC1L1fwGTJXmGEb/wIDtVeAVLMsyQAA\nAABJRU5ErkJggg==\n",
       "prompt_number": 133,
       "text": "\n     1     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  ___   ___\n\u2572\u2571 \u03c0 \u22c5\u2572\u2571 t "
      }
     ],
     "prompt_number": 133
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 134,
       "text": "\n\u239b  1           \u239e\n\u239c\u2500\u2500\u2500\u2500\u2500, 0, True\u239f\n\u239c  ___         \u239f\n\u239d\u2572\u2571 p          \u23a0"
      }
     ],
     "prompt_number": 134
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAAuBAMAAADZ8SrdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfwAPv80ODas4hwAAAABJRU5ErkJggg==\n",
       "prompt_number": 135,
       "text": "\n                       \u239b  1              \u239e\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239c  ___            \u239f\n                       \u239d\u2572\u2571 p             \u23a0"
      }
     ],
     "prompt_number": 135
    },
    {
     "cell_type": "markdown",
     "source": "34"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2*sqrt(t/pi)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACYAAAAtBAMAAAAw1AReAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZomZu1R2IjLvq90Q\nRM0XT9VLAAABQ0lEQVQoFY2SL0/DQBiHf4x2vXXNBmjEJIIEBJnuDAQD+wgNIUERqqeaYJcgENjK\n4cAjKvCQkEyQED4ABhL+uePtdXf39mp4xd3vfe7aXnMPwEtIKd85oOwVDqD2vImwhyBzsMjhDRwW\npug/OmwCXDgIMzy9DeswyIFdhsp+OQN+GZt/AfeAqP3FAbABLMVsH16ziPow8Rnsxi06Wm+8zpj/\nQ6+DP8wYw+Wp7dZ2xqo5iw1rp+JDNQ/VUpk7Ca4VE2pUQ3eArcK2Ou3rYOfo22advG2d7HwE9OkO\nbdHRk8iuL9IcbZeJ0eqxZqUGVOhI+anZ/zXQT5iZNGgUadCoCaJD+mxtZYZwuvlS8L107dNg5cQg\nrUErzQ1baIAebgxDpQGecWVZpQFGuLWs0gB3YP5VGtg9KnEN9BLTQCMwDQxjGvwBm6tWnGldLKsA\nAAAASUVORK5CYII=\n",
       "prompt_number": 136,
       "text": "\n    ___\n2\u22c5\u2572\u2571 t \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   ___ \n \u2572\u2571 \u03c0  "
      }
     ],
     "prompt_number": 136
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 137,
       "text": "\n\u239b 1           \u239e\n\u239c\u2500\u2500\u2500\u2500, 0, True\u239f\n\u239c 3/2         \u239f\n\u239dp            \u23a0"
      }
     ],
     "prompt_number": 137
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIcAAAAtBAMAAAB2TmWxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nvo2aIHptjn9QOBUpJoMlsQAAAABJRU5ErkJggg==\n",
       "prompt_number": 138,
       "text": "\n    ___             \n2\u22c5\u2572\u2571 t \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         ___        \n       \u2572\u2571 \u03c0         "
      }
     ],
     "prompt_number": 138
    },
    {
     "cell_type": "markdown",
     "source": "35"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "factorial(n)/factorial(2*n)*4**n/sqrt(pi)*t**(n-S(1)/2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAyBAMAAAANeoDTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 139,
       "text": "\n n  n - 1/2   \n4 \u22c5t       \u22c5n!\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   ___        \n \u2572\u2571 \u03c0 \u22c5(2\u22c5n)! "
      }
     ],
     "prompt_number": 139
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 140,
       "text": "\n\u239b n  -n - 1/2                               \u239e\n\u239c4 \u22c5p        \u22c5n!\u22c5(n - 1/2)!                 \u239f\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, -n + 1/2 < 1\u239f\n\u239c         ___                               \u239f\n\u239d       \u2572\u2571 \u03c0 \u22c5(2\u22c5n)!                        \u23a0"
      }
     ],
     "prompt_number": 140
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simplify(_[0])",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ0AAAAzBAMAAAB/Hhd7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nANkdMgb6oXKRAAAAAElFTkSuQmCC\n",
       "prompt_number": 141,
       "text": "\n n  -n - 1/2              \n4 \u22c5p        \u22c5n!\u22c5(n - 1/2)!\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         ___              \n       \u2572\u2571 \u03c0 \u22c5(2\u22c5n)!       "
      }
     ],
     "prompt_number": 141
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "combsimp(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ0AAAAzBAMAAAB/Hhd7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIqvNu3YyRJkQZlSJ\n3e/ayhD9AAAD+UlEQVRIDc1WXYgbVRg9k83O7EySzaAi0pekD9YfEHwq9G3EiqIPpsWirQuNWtBC\n3QSlWEHsUhC2rWAQV6QimYoF3Rfz4PqilrwoFlpZHyt9iA/SLgpmH0rrD6Tnu3Nn9yYZmxkR7Ae5\n93znO/fkzs1M5gNuHoduXs5a9e7NumKC/siEetby/+GX99PvcnR/d44vLfzgw22O84nMiJ9VTVDN\ncn/PJvAx5e7eMqex80AYs2rOt4bSKBE/o2BVDQNRFEq18tgy+7JQe8Z4EuJnVwkizVSwaeAtLi6e\nQBsvI0KDwaAfWXh/ynwxSoZH8cOP/ESa3YEyMDRf4LzOCs+EGjl/ERQCnQ1Nyk9OSGnsIwE2DZTw\nLWyvRSuW8LNeal8jyHV1NjQpvzdJKc30TABlsHJ8+Xik24pvmxF6Ho8CZ87tOwlrnUypCW9u6dNO\nVNSjs/9+onYLkaZGPzFwP7uKxSEhExdrcE//jXngD6ZlH6cqXbs3KmM+KzugxqnTT8J5v48LCplD\nMYTz4Rq+AfqkKy28frZVWDUVGpdCAmpOQfthOsDOMeFXZGaqOAY8SNTg55jkwB07GY8QgPfAAJiq\nEVLz9fkLl5pCIx/iNwWMwem8yFML8SuwQFr8fsJsx1DEUPktSJYLODDKvrMWoc2xMuCRtH27/xx8\nsrxedx0NW/BIlORLFJ8PolKjrrgRHdOzPIqutZ2o7aPQw1PFIY29Zxtz+T2Uxn7nel3VP1n+bki3\nkczBOuGrZ4mLpru46/RGScAS5lvqp9fPW1y8Owb/POdrCbUGytwb76iRuDySJ6ReNYG05Rjw8WjF\nkht2Qri9RMHbZOV4h+LV6zyFSXEySeAt8NFdTapM5PJJisNoYaaeVJnIOd1xib13Sx1vjPOpmC/H\nVRU+bm5tnL81Gfl3+O+if2teZMpd2dFBUO0ZK0IDZ4ObL/2txsIPDJwNvhTLvQ6875/QWUHutflm\nXMswPxRrS8AuNEKd7uD8eZxoLs1kV2PVCvALcj2d7uN8tBPX0s85P9beA1xCMf5De4F0419c721w\nn+QvTNe+GOd7ujeQ5quy8V1SShfbkDt47rUWuwTl167p3mCWVJmfjGFVcdAKd3GV1ZOlj8W9Qcmn\nnzDpw6qyLagDRZ8A1ioHvth0b1Cqsx0gkyEOs9t7j/opiJV6bazwXzjqDeR6ZzKYifRhQN7Gr0T9\nX5/tUg013RvItVoiyhBH625A+ePcEoMvrwPLZwLdGxwgc3td+PRRCopNqi+y1WSwDf99MAh4hNIb\nLJPJ+rwV1uX44vgoBmqW5y1zXHnaWKL2GudON0ZZ5v2Bqb7PSN41cHrY7pjaJSPZa+D00E4vNZU3\nANkdMgb6oXKRAAAAAElFTkSuQmCC\n",
       "prompt_number": 142,
       "text": "\n n  -n - 1/2              \n4 \u22c5p        \u22c5n!\u22c5(n - 1/2)!\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         ___              \n       \u2572\u2571 \u03c0 \u22c5(2\u22c5n)!       "
      }
     ],
     "prompt_number": 142
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAK4AAAAyBAMAAAA3hv3lAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJhBO5DGZifndTSB2jKlD9x++EXwLo6i2EgAAAABJRU5ErkJggg==\n",
       "prompt_number": 143,
       "text": "\n n  n - 1/2                \n4 \u22c5t       \u22c5Heaviside(t)\u22c5n!\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          ___              \n        \u2572\u2571 \u03c0 \u22c5(2\u22c5n)!       "
      }
     ],
     "prompt_number": 143
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/(p**n*sqrt(p))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAAsBAMAAABvbn4JAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHaZEGZEMondu+8i\nq80NNKsLAAAA+UlEQVQoFWNggAMW3aMacA6IwcipwA8TYDU2NjZl8GeohAlAaCmGDlQBI4bnCkAR\nIZNQE4jEHYamBAYGlpBPDMYQAQjJ5v6BoRNZgIFpAcMqFAGuAIYdKAL8B9i2oAjUF3AKoAhEhbaj\n8BkuoXIZGLahCfB8QRPI++yAJkI6l/8/CvhAugl4dLDXgwxHMpPLAU11MRqfYQ2aAPsFNAHeA0gx\nBpI7ix5j1xlQY4wHaAQ8xkAc5gIGBniMJX1kYPACGoOIsXUMDKCoQcRYfgHLAqAAIsY4F3AkAAUQ\nMcb4B2QEcoxtVgPykWNMfwFQADnG/AWAAsiAHYkDAELpR0aKkXVcAAAAAElFTkSuQmCC\n",
       "prompt_number": 144,
       "text": "\n  -n \n p   \n\u2500\u2500\u2500\u2500\u2500\n  ___\n\u2572\u2571 p "
      }
     ],
     "prompt_number": 144
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIwAAAA1BAMAAABhKx6wAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 145,
       "text": "\n n - 1/2             \nt       \u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n      \u0393(n + 1/2)     "
      }
     ],
     "prompt_number": 145
    },
    {
     "cell_type": "markdown",
     "source": "36"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/sqrt(pi*t)*exp(-a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAAvBAMAAABXrFT6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDLN\n3atI+RUPAAABXUlEQVQ4EdWSv0vDQBiG3zZJm1BS01b34C8EB60ZdYibIqiToJODUP+MriJIQCpu\nBpzV/AFiO7uKILgUcXPpYEF0qF+a3N1HaEcH3+Hueb43cBc4IBsryk5S15wxhe2PKqbnNhoLIwrN\nvcJtZl7zPM/ZCe5RzxRDvcE8Xq0RzRlenN4pK6o1f2haZSash6owDnVXGaMGCj5Thf3qpBJG+hcT\njtYnEPCB5APkmlI4nFdmuf5LHozJX/9Mm84VZ8Q8GPQS3RdT2jmXllXBGYavCs54ALZERazyDuwJ\nI5bRu4B4HjHLlCOU0uuBuEW3pddC2Ya59uQOkRgfF5cn8RBYBYx4nnApWM/7MJdCWLvARATJOCrT\nV0YHBRq2CQWjY5OYP2jRNhUXKZvdok/2jGNa3xBKNhzbIbnbpCPwmAsk24HmkhRXmrReL9KSckyU\nfD/ZswzrWxWcAbqsDGc5/AWXaGjClzkvzAAAAABJRU5ErkJggg==\n",
       "prompt_number": 146,
       "text": "\n    -a\u22c5t   \n   \u212f       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  ___   ___\n\u2572\u2571 \u03c0 \u22c5\u2572\u2571 t "
      }
     ],
     "prompt_number": 146
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 147,
       "text": "\n\u239b        1                                         \u03c0\u239e\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument(a, \u221e)\u2502 < \u2500\u239f\n\u239c          _______                                 2\u239f\n\u239c  ___    \u2571 a + p                                   \u239f\n\u239c\u2572\u2571 a \u22c5  \u2571  \u2500\u2500\u2500\u2500\u2500                                   \u239f\n\u239d      \u2572\u2571     a                                     \u23a0"
      }
     ],
     "prompt_number": 147
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#radsimp(_[0])\n# Takes forever",
     "language": "python",
     "outputs": [],
     "prompt_number": 148
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t) ",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAABEBAMAAAC17JxwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSGC+sYIEnoBeXi/5A51nRTLv3dmQAAAAAElFTkSuQmCC\n",
       "prompt_number": 149,
       "text": "\n                       \u239b     1                 \u239e\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239c    _______            \u239f\n                       \u239c   \u2571     p             \u239f\n                       \u239c  \u2571  1 + \u2500             \u239f\n                       \u239d\u2572\u2571       a             \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                       ___                      \n                     \u2572\u2571 a                       "
      }
     ],
     "prompt_number": 149
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/sqrt(p+a)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAqBAMAAAD7dO6DAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73ZnN\n72bDLMX7AAABNklEQVQ4EWNgQAaMysg8NDZL2Hc0EVTuqCwiPDgif21G8IYn6z8e8IH2XnbBZj3c\n3on4HMCyAZ8sZwNclqUAzoQxnGAMBgb2Bwg2lKWBEGF/gGBDWBwKDAxseVVgDvsDMFW28d1GMIOB\ngcmBgWEawy4wj/0BiGJ8+I1hN4gBBE0MDHwLoDz2ByARZpEPDFFAGijDsIWBgTeAYTmQtXPmnJUz\nJwEZbBMYZgIV5RcwME5gYOg3YPwIFIS7mfMBw1Ugj/MAA6sDA4O/AN8CsE/ZHwAFGRj4G5gvAinm\nLwySQIq3oUchAUgzsD8AEkDVDlxgxSsZgoE85n0P1RpAwuwPgAQDw8t3IWC6vhFoLQIwg7UwqEFF\neLcZIOTgrMtQFttPuBCCwQHxADBYsBU0Nr8FoEodEFrQWAASQHXBLHYbfwAAAABJRU5ErkJggg==\n",
       "prompt_number": 150,
       "text": "\n    1    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  _______\n\u2572\u2571 a + p "
      }
     ],
     "prompt_number": 150
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIkAAAAsBAMAAACj24anAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVM2ruzLvEImZInbd\nZkTWwhNwAAACx0lEQVRIDe2WTWgTQRTH/5uvNZt0s1cRaaEHL4I9K4WoAQUPXVTEk/FW9SALqQZP\nycWiINKDUkIuAcEP8JCDICiSHAqlF01BqIUGQuil4CGCJUjF+GZ2szPpbmRTIl58sDP/ee/NL/Mm\nM8kCXsv29rxO2TPX68hDf932d8veAJRtOZ/pXGu/5wCUqzVjDBSk/lPYNnq+o3+yL5HKGNaiLI6D\ngthfpLycW2uxSiVzbkCDuXgDKCdmnIQha5GmO9KmhCtseM/2nb7SxWapVGoMq8hOk1ubsg5oJpQq\ni4SL8XknY8S11IFIETgGsklLYx2z0ShsEakW8IFNnWYNN702y3rP2bWjcssrCreALHkv0qN9k8NM\nB6WkgJ3jZVoPzUl0D0op0MQL9MQsIJQmMWBsLb0/GJXDK3pOebv0JCtAoUpiwIJW9BqIMlzSBGrW\nAIIGQSl3gYk05bOK1qj/Qo+woBTa1XBV4bs70fs5e21VfXP0qYuRKFm/7aFEVghCJkLGBnAZeJU/\n/OmmpSQN9q3bJlHO9X37ek5JVKGUTaDpBmP46mpBiS65zkHBKWrddvIbwOU2Hrp5gmL/9vFA1HDj\nJDgFj7lLL7qRB+qqqwXlkuuD0hC6TwlbzJcTgfObFXcgKCddny9FrbL4W5F0XUhxXrQZ4P16nYec\ntRxZbi7TuCOlCxmdElpQwiYi1Xidh2yKurGHEo39KQsf/SjPgC3ElySKfqeDR0MpEsO5AUQAHcQb\nSBqknmTOrGTOkkikkaHOfy0UEMZ2V6d3FjUN7RcKFo/YFSHSIHBgCiJTOGQi2sUW3RQyh5Kq6PMj\nUPRdLFD6LW06wSB9StbkFQasCFjhR3OnmXsnUz4377NhYMrkizSf7jS6wcUpexiYEiu3HIDc3R6R\nkvguz3a01g1MaectylU9/zDkbP/goXyAipw3eNP5fG/neYP/DR2uAyX9ZP8OAAAAAElFTkSuQmCC\n",
       "prompt_number": 151,
       "text": "\n                       \u239b    1                \u239e\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239c  _______            \u239f\n                       \u239d\u2572\u2571 a + p             \u23a0"
      }
     ],
     "prompt_number": 151
    },
    {
     "cell_type": "markdown",
     "source": "37"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/(2*t*sqrt(pi*t))*(exp(-b*t)-exp(-a*t))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFcAAAAyBAMAAADb31q2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nv6H1nXhz/TdsAAAAAElFTkSuQmCC\n",
       "prompt_number": 152,
       "text": "\n -b\u22c5t    -a\u22c5t\n\u212f     - \u212f    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     ___  3/2\n 2\u22c5\u2572\u2571 \u03c0 \u22c5t   "
      }
     ],
     "prompt_number": 152
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [],
     "prompt_number": 153
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt(p+a)-sqrt(p+b)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIcAAAAWBAMAAAAY85LfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMrsQmSKrRHZUzYnd\n72ZiHOLWAAACF0lEQVQ4EZVTv2sTURz/XH5danKXG7rokuPEUjKU1D+gDdZR5HDIoEPiYOlmoEsV\nSg4RShE0o4NI/gBBnXRwCHbqJiiIiBCsmxBTbKAupt/33vdy19xrId/h3efHe5/7vm8uwGxVHidr\nOFsEcHnWA5r9Rk0jhpKRbHI81rSZ88MTmmeqqhE1Uh2XDjWykubPdE4b92Bo+uM9V6K9FyOYQNkm\n7K5Sjc60azQj5WUEEyjjIFdT6lx/2o3P67yQq4DFHSRD6kD+6QsVrULmHq/8Sg77HdAYvJf7OOT5\n2mCNW7oLLOJGPGRlNcjU2J080i7wp2U5QlAh6YNjXFe+mFcvJKqTzTIKPeWq9Q49Ci1gHcVAKCrE\nXB7iA0ABoHlZb7BPVsbzXnkeSXiElAvc8qgWiZr36XIXCHxDqQ9c8yofvQWieRcesPUPoHk1Wum/\npFHxYJdEblTZErFdID1C2xcyzyTVx3diFeAtUK7avY48wyEjbFcl56XwCehSyJAuJIpDSr75mVjb\nEfOy/NvNgFjYiXEorycVuWT/w+4T6pl7knNI2SmKdxddMS/z5sGGL13Vib3025F0slTkSDD/pCol\nDvk5oGtQXyMaSaweSJwLYpKEbWcQl8yOZBusfdmJm4yt1rRYbHanJeJfWXvmasyGajrm2MdBjDHM\n8k+KVdXYqR3m6/BjjuQfEZygh0f8MmMinQ/Uf0a35wT0Q45E5HZmzwAAAABJRU5ErkJggg==\n",
       "prompt_number": 154,
       "text": "\n  _______     _______\n\u2572\u2571 a + p  - \u2572\u2571 b + p "
      }
     ],
     "prompt_number": 154
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAARwAAAAhBAMAAAD5SQkUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
       "prompt_number": 155,
       "text": "\n                       \u239b  _______            \u239e                          \u239b  ___\nInverseLaplaceTransform\u239d\u2572\u2571 a + p , p, t, None\u23a0 - InverseLaplaceTransform\u239d\u2572\u2571 b \n\n____            \u239e\n+ p , p, t, None\u23a0"
      }
     ],
     "prompt_number": 155
    },
    {
     "cell_type": "markdown",
     "source": "38"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sin(a*t)/(t*sqrt(2*pi*t))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAyBAMAAADSNPrMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWCeoYwE6fs3xAB33WuOtQecvwmvRjxr4tkoAAAAASUVORK5CYII=\n",
       "prompt_number": 156,
       "text": "\n  ___         \n\u2572\u2571 2 \u22c5sin(a\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     ___  3/2 \n 2\u22c5\u2572\u2571 \u03c0 \u22c5t    "
      }
     ],
     "prompt_number": 156
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 157,
       "text": "\n\u239b                a                     \u2502                 \u239b          2      \u239e\u2502 \n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 \n\u239c             ____________________                                            \n\u239c            \u2571       ________                                                 \n\u239c           \u2571       \u2571  2                                                      \n\u239c  ___     \u2571       \u2571  a                                                       \n\u239c\u2572\u2571 p \u22c5   \u2571       \u2571   \u2500\u2500 + 1  + 1                                             \n\u239c        \u2571       \u2571     2                                                      \n\u239d      \u2572\u2571      \u2572\u2571     p                                                       \n\n   \u239e\n= 0\u239f\n   \u239f\n   \u239f\n   \u239f\n   \u239f\n   \u239f\n   \u239f\n   \u23a0"
      }
     ],
     "prompt_number": 157
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAO0AAABbBAMAAABt8B/UAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 158,
       "text": "\n                         \u239b                1                            \u239e\na\u22c5InverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                         \u239c             ____________________            \u239f\n                         \u239c            \u2571       ________                 \u239f\n                         \u239c           \u2571       \u2571  2                      \u239f\n                         \u239c  ___     \u2571       \u2571  a                       \u239f\n                         \u239c\u2572\u2571 p \u22c5   \u2571       \u2571   \u2500\u2500 + 1  + 1             \u239f\n                         \u239c        \u2571       \u2571     2                      \u239f\n                         \u239d      \u2572\u2571      \u2572\u2571     p                       \u23a0"
      }
     ],
     "prompt_number": 158
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt(sqrt(p**2+a**2)-p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIoAAAAhBAMAAAD0ci56AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nyif/Tn+s+sKc73j3se3c4i+faN7OZmDpkQAAAABJRU5ErkJggg==\n",
       "prompt_number": 159,
       "text": "\n    ___________________\n   \u2571         _________ \n  \u2571         \u2571  2    2  \n\u2572\u2571   -p + \u2572\u2571  a  + p   "
      }
     ],
     "prompt_number": 159
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANcAAAAsBAMAAAAEBjBcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 160,
       "text": "\n                       \u239b    ___________________            \u239e\n                       \u239c   \u2571         _________             \u239f\n                       \u239c  \u2571         \u2571  2    2              \u239f\nInverseLaplaceTransform\u239d\u2572\u2571   -p + \u2572\u2571  a  + p   , p, t, None\u23a0"
      }
     ],
     "prompt_number": 160
    },
    {
     "cell_type": "markdown",
     "source": "39"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt(2/(pi*t))*sin(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHEAAAA5BAMAAADzW9niAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 161,
       "text": "\n          ___         \n  ___    \u2571 1          \n\u2572\u2571 2 \u22c5  \u2571  \u2500 \u22c5sin(a\u22c5t)\n      \u2572\u2571   t          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          ___         \n        \u2572\u2571 \u03c0          "
      }
     ],
     "prompt_number": 161
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 162,
       "text": "\n\u239b     \u239b              ________     \u239e                                           \n\u239c     \u239c             \u2571  2          \u239f                                           \n\u239c     \u239c 2    2     \u2571  a          2\u239f                                           \n\u239c   a\u22c5\u239ca  + p \u22c5   \u2571   \u2500\u2500 + 1  + p \u239f                                           \n\u239c     \u239c          \u2571     2          \u239f                                           \n\u239c     \u239d        \u2572\u2571     p           \u23a0         \u2502                 \u239b          2    \n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a),\n\u239c                                   3/2                                       \n\u239c               \u239b      ________    \u239e                                          \n\u239c               \u239c     \u2571  2         \u239f                                          \n\u239c 3/2 \u239b 2    2\u239e \u239c    \u2571  a          \u239f                                          \n\u239cp   \u22c5\u239da  + p \u23a0\u22c5\u239c   \u2571   \u2500\u2500 + 1  + 1\u239f                                          \n\u239c               \u239c  \u2571     2         \u239f                                          \n\u239d               \u239d\u2572\u2571     p          \u23a0                                          \n\n        \u239e\n        \u239f\n        \u239f\n        \u239f\n        \u239f\n  \u239e\u2502    \u239f\n \u221e\u23a0\u2502 = 0\u239f\n        \u239f\n        \u239f\n        \u239f\n        \u239f\n        \u239f\n        \u239f\n        \u23a0"
      }
     ],
     "prompt_number": 162
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABVwAAABYBAMAAAD7ByE/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8TFss6L8P1knBKHtSuSfAAAAAElFTkSuQmCC\n",
       "prompt_number": 163,
       "text": "\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n 3                        \u239b                              1                    \na \u22c5InverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                          \u239c                            3/2                    \n                          \u239c        \u239b      ________    \u239e           \u239b      _____\n                          \u239c        \u239c     \u2571  2         \u239f           \u239c     \u2571  2  \n                          \u239c 2  3/2 \u239c    \u2571  a          \u239f       7/2 \u239c    \u2571  a   \n                          \u239ca \u22c5p   \u22c5\u239c   \u2571   \u2500\u2500 + 1  + 1\u239f    + p   \u22c5\u239c   \u2571   \u2500\u2500 +\n                          \u239c        \u239c  \u2571     2         \u239f           \u239c  \u2571     2  \n                          \u239d        \u239d\u2572\u2571     p          \u23a0           \u239d\u2572\u2571     p   \n\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                    \u239b                         \n                       \u239e                            \u239c                         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f + a\u22c5InverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        3/2            \u239f                            \u239c                         \n___    \u239e               \u239f                            \u239c        \u239b      ________  \n       \u239f               \u239f                            \u239c        \u239c     \u2571  2       \n       \u239f               \u239f                            \u239c 2  3/2 \u239c    \u2571  a        \n 1  + 1\u239f               \u239f                            \u239ca \u22c5p   \u22c5\u239c   \u2571   \u2500\u2500 + 1  +\n       \u239f               \u239f                            \u239c        \u239c  \u2571     2       \n       \u23a0               \u23a0                            \u239d        \u239d\u2572\u2571     p        \n\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n      2                                          \u239e                            \n     p                                           \u239f                            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f + a\u22c5InverseLaplaceTransform\n   3/2                            3/2            \u239f                            \n  \u239e           \u239b      ________    \u239e               \u239f                            \n  \u239f           \u239c     \u2571  2         \u239f               \u239f                            \n  \u239f       7/2 \u239c    \u2571  a          \u239f               \u239f                            \n 1\u239f    + p   \u22c5\u239c   \u2571   \u2500\u2500 + 1  + 1\u239f               \u239f                            \n  \u239f           \u239c  \u2571     2         \u239f               \u239f                            \n  \u23a0           \u239d\u2572\u2571     p          \u23a0               \u23a0                            \n\n\u239b                               ________                                   \u239e\n\u239c                              \u2571  2                                        \u239f\n\u239c                       2     \u2571  a                                         \u239f\n\u239c                      p \u22c5   \u2571   \u2500\u2500 + 1                                    \u239f\n\u239c                           \u2571     2                                        \u239f\n\u239c                         \u2572\u2571     p                                         \u239f\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n\u239c                            3/2                            3/2            \u239f\n\u239c        \u239b      ________    \u239e           \u239b      ________    \u239e               \u239f\n\u239c        \u239c     \u2571  2         \u239f           \u239c     \u2571  2         \u239f               \u239f\n\u239c 2  3/2 \u239c    \u2571  a          \u239f       7/2 \u239c    \u2571  a          \u239f               \u239f\n\u239ca \u22c5p   \u22c5\u239c   \u2571   \u2500\u2500 + 1  + 1\u239f    + p   \u22c5\u239c   \u2571   \u2500\u2500 + 1  + 1\u239f               \u239f\n\u239c        \u239c  \u2571     2         \u239f           \u239c  \u2571     2         \u239f               \u239f\n\u239d        \u239d\u2572\u2571     p          \u23a0           \u239d\u2572\u2571     p          \u23a0               \u23a0"
      }
     ],
     "prompt_number": 163
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt((sqrt(p**2+a**2)-p)/(p**2+a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAI0AAAA3BAMAAADDIdSFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 164,
       "text": "\n       ___________________\n      \u2571         _________ \n     \u2571         \u2571  2    2  \n    \u2571   -p + \u2572\u2571  a  + p   \n   \u2571    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \n  \u2571           2    2      \n\u2572\u2571           a  + p       "
      }
     ],
     "prompt_number": 164
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAREAAAA4BAMAAAAobtH3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 165,
       "text": "\n                       \u239b      __________________________            \u239e\n                       \u239c     \u2571      p           1                   \u239f\nInverseLaplaceTransform\u239c    \u2571  - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 , p, t, None\u239f\n                       \u239c   \u2571      2    2      _________             \u239f\n                       \u239c  \u2571      a  + p      \u2571  2    2              \u239f\n                       \u239d\u2572\u2571                 \u2572\u2571  a  + p               \u23a0"
      }
     ],
     "prompt_number": 165
    },
    {
     "cell_type": "markdown",
     "source": "40"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt(2/(pi*t))*cos(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHMAAAA5BAMAAAD3rgnfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nCRhc9nfdYGYDWtd1G9DgQiYYBv4D32XqrENqnxkAAAAASUVORK5CYII=\n",
       "prompt_number": 166,
       "text": "\n          ___         \n  ___    \u2571 1          \n\u2572\u2571 2 \u22c5  \u2571  \u2500 \u22c5cos(a\u22c5t)\n      \u2572\u2571   t          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          ___         \n        \u2572\u2571 \u03c0          "
      }
     ],
     "prompt_number": 166
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 167,
       "text": "\n\u239b          \u239b    \u239ba\u239e\u239e                                                \u239e\n\u239c          \u239catan\u239c\u2500\u239f\u239f                                                \u239f\n\u239c   ___    \u239c    \u239dp\u23a0\u239f                                                \u239f\n\u239c \u2572\u2571 2 \u22c5cos\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                                                \u239f\n\u239c          \u239d   2   \u23a0      \u2502                 \u239b          2      \u239e\u2502    \u239f\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 = 0\u239f\n\u239c            ________                                               \u239f\n\u239c           \u2571  2                                                    \u239f\n\u239c  ___     \u2571  a                                                     \u239f\n\u239c\u2572\u2571 p \u22c5   \u2571   \u2500\u2500 + 1                                                \u239f\n\u239c      4 \u2571     2                                                    \u239f\n\u239d      \u2572\u2571     p                                                     \u23a0"
      }
     ],
     "prompt_number": 167
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPQAAABICAMAAAD2+SjCAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nHQGP/+vRHZQvaxsvMP621P8BnD5MsSBiei0AAAAASUVORK5CYII=\n",
       "prompt_number": 168,
       "text": "\n                             \u239b       \u239b    \u239ba\u239e\u239e                \u239e\n                             \u239c       \u239catan\u239c\u2500\u239f\u239f                \u239f\n                             \u239c       \u239c    \u239dp\u23a0\u239f                \u239f\n                             \u239c    cos\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                \u239f\n  ___                        \u239c       \u239d   2   \u23a0                \u239f\n\u2572\u2571 2 \u22c5InverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                             \u239c            ________            \u239f\n                             \u239c           \u2571  2                 \u239f\n                             \u239c  ___     \u2571  a                  \u239f\n                             \u239c\u2572\u2571 p \u22c5   \u2571   \u2500\u2500 + 1             \u239f\n                             \u239c      4 \u2571     2                 \u239f\n                             \u239d      \u2572\u2571     p                  \u23a0"
      }
     ],
     "prompt_number": 168
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt((sqrt(p**2+a**2)+p)/(p**2+a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAA3BAMAAADXmIghAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nh0N/AVl4PVxK8kO3AAAAAElFTkSuQmCC\n",
       "prompt_number": 169,
       "text": "\n       __________________\n      \u2571        _________ \n     \u2571        \u2571  2    2  \n    \u2571   p + \u2572\u2571  a  + p   \n   \u2571    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \n  \u2571          2    2      \n\u2572\u2571          a  + p       "
      }
     ],
     "prompt_number": 169
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQMAAAA4BAMAAAALNYAiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n4CD+x+QGRuxx5cF79JGKJyO1ySv/AxWi5cUfSoFdAAAAAElFTkSuQmCC\n",
       "prompt_number": 170,
       "text": "\n                       \u239b      ________________________            \u239e\n                       \u239c     \u2571    p           1                   \u239f\nInverseLaplaceTransform\u239c    \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 , p, t, None\u239f\n                       \u239c   \u2571    2    2      _________             \u239f\n                       \u239c  \u2571    a  + p      \u2571  2    2              \u239f\n                       \u239d\u2572\u2571               \u2572\u2571  a  + p               \u23a0"
      }
     ],
     "prompt_number": 170
    },
    {
     "cell_type": "markdown",
     "source": "41"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt(2/(pi*t))*sinh(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAA5BAMAAADkeUkrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 171,
       "text": "\n          ___          \n  ___    \u2571 1           \n\u2572\u2571 2 \u22c5  \u2571  \u2500 \u22c5sinh(a\u22c5t)\n      \u2572\u2571   t           \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n           ___         \n         \u2572\u2571 \u03c0          "
      }
     ],
     "prompt_number": 171
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#laplace_transform(_, t, p)\n# Takes forever",
     "language": "python",
     "outputs": [],
     "prompt_number": 172
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt((sqrt(p**2-a**2)-p)/(p**2-a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJsAAAA3BAMAAADpkSUqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nUccLc//L/AWKQmlsilSvxQAAAABJRU5ErkJggg==\n",
       "prompt_number": 173,
       "text": "\n       _____________________\n      \u2571         ___________ \n     \u2571         \u2571    2    2  \n    \u2571   -p + \u2572\u2571  - a  + p   \n   \u2571    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \n  \u2571             2    2      \n\u2572\u2571           - a  + p       "
      }
     ],
     "prompt_number": 173
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAS0AAAA4BAMAAABaobNBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nU6oX/r7lNwXiGU3moP6R4HB9R/8B9rkdO8XtoB4AAAAASUVORK5CYII=\n",
       "prompt_number": 174,
       "text": "\n                       \u239b      ______________________________            \u239e\n                       \u239c     \u2571       p             1                    \u239f\nInverseLaplaceTransform\u239c    \u2571  - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 , p, t, None\u239f\n                       \u239c   \u2571        2    2      ___________             \u239f\n                       \u239c  \u2571      - a  + p      \u2571    2    2              \u239f\n                       \u239d\u2572\u2571                   \u2572\u2571  - a  + p               \u23a0"
      }
     ],
     "prompt_number": 174
    },
    {
     "cell_type": "markdown",
     "source": "42"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt(2/(pi*t))*cosh(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAA5BAMAAADpZzlsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n4wsB6H8ATyv8Sz2MNuAAAAAASUVORK5CYII=\n",
       "prompt_number": 175,
       "text": "\n          ___          \n  ___    \u2571 1           \n\u2572\u2571 2 \u22c5  \u2571  \u2500 \u22c5cosh(a\u22c5t)\n      \u2572\u2571   t           \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n           ___         \n         \u2572\u2571 \u03c0          "
      }
     ],
     "prompt_number": 175
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 176,
       "text": "\n\u239b\u23a7      \u239b                                                  \u239b       \u239b          \n\u239c\u23aa      \u239c                                                  \u239c   ___ \u239c         4\n\u239c\u23aa      \u239c                                                  \u239c \u2572\u2571 2 \u22c5\u239c1 - \u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23aa      \u239c                                                  \u239c       \u239c       \u239b  \n\u239c\u23aa      \u239c                                                  \u239c       \u239c     2 \u239c  \n\u239c\u23aa      \u239c                                                  \u239c       \u239c    a \u22c5\u239c-4\n\u239c\u23aa      \u239c                                                  \u239c       \u239c       \u239c  \n\u239c\u23aa      \u239c                                            ___   \u239c       \u239d       \u239d  \n\u239c\u23aa      \u239c                                        2\u22c5\u2572\u2571 \u03c0 \u22c5p\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23aa      \u239c                                                  \u239c       ___________\n\u239c\u23aa      \u239c  ___ \u239b      1               1      \u239e             \u239c      \u2571       ____\n\u239c\u23aa      \u239c\u2572\u2571 \u03c0 \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f             \u239c     \u2571       \u2571    \n\u239c\u23aa      \u239c      \u239c      _______         _______\u239f             \u239c    \u2571       \u2571     \n\u239c\u23aa      \u239c      \u239c     \u2571     p         \u2571     p \u239f             \u239c   \u2571       \u2571   1 -\n\u239c\u23aa      \u239c      \u239c2\u22c5  \u2571  1 + \u2500    2\u22c5  \u2571  1 - \u2500 \u239f             \u239c  \u2571       \u2571       \n\u239c\u23aa  ___ \u239c      \u239d  \u2572\u2571       a      \u2572\u2571       a \u23a0             \u239d\u2572\u2571      \u2572\u2571        \n\u239c\u23aa\u2572\u2571 \u03c0 \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23aa      \u239c            \u0393(1/4)\u22c5\u0393(3/4)                                            \n\u239c\u23aa      \u239d                                                                     \n\u239c\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23a8                                                                   ___      \n\u239c\u23aa                                                                 \u2572\u2571 a       \n\u239c\u23aa                                                                            \n\u239c\u23aa                                            \u239b      1                1       \n\u239c\u23aa                                            \u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23aa                                            \u239c      _______         _________\n\u239c\u23aa                                            \u239c     \u2571 a             \u2571   a     \n\u239c\u23aa                                            \u239c2\u22c5  \u2571  \u2500 + 1    2\u22c5  \u2571  - \u2500 + 1 \n\u239c\u23aa                                            \u239d  \u2572\u2571   p          \u2572\u2571     p     \n\u239c\u23aa                                            \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23aa                                                                    ___     \n\u239c\u23aa                                                                \u03c0\u22c5\u2572\u2571 p      \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                               \u239b            \n\u239c\u23aa                                                   ___ \u256d\u2500\u256e2, 1 \u239c 3/4    1/4,\n\u239c\u23aa                                                 \u2572\u2571 \u03c0 \u22c5\u2502\u2576\u2510     \u239c            \n\u239c\u23aa                                                       \u2570\u2500\u256f3, 3 \u239c0, 1/2    1/\n\u239c\u23aa                                                               \u239d            \n\u239c\u23aa                                                 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23aa                                                                   ___      \n\u239d\u23a9                                                                 \u2572\u2571 a       \n\n  2     \u239e                                         \u239e       \u239e                   \n\u22c5p      \u239f                                         \u239f       \u239f                   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                                         \u239f       \u239f                   \n      2\u239e\u239f                                         \u239f       \u239f                   \n   4\u22c5p \u239f\u239f                                         \u239f       \u239f                   \n + \u2500\u2500\u2500\u2500\u239f\u239f                                         \u239f       \u239f                   \n     2 \u239f\u239f                     ___  2              \u239f       \u239f                   \n    a  \u23a0\u23a0                 4\u22c5\u2572\u2571 2 \u22c5p               \u239f       \u239f                   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u0393(3/4)\u239f                   \n_________                                      3/2\u239f       \u239f                   \n____                       \u239b      ________    \u239e   \u239f       \u239f                   \n  2            \u239b        2\u239e \u239c     \u2571      2     \u239f   \u239f       \u239f                   \n p           2 \u239c     4\u22c5p \u239f \u239c    \u2571      p      \u239f   \u239f       \u239f                   \n \u2500\u2500  + 1    a \u22c5\u239c-4 + \u2500\u2500\u2500\u2500\u239f\u22c5\u239c   \u2571   1 - \u2500\u2500  + 1\u239f   \u239f       \u239f                   \n  2            \u239c       2 \u239f \u239c  \u2571         2     \u239f   \u239f       \u239f                   \n a             \u239d      a  \u23a0 \u239d\u2572\u2571         a      \u23a0   \u23a0       \u239f                   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                   \n               2                                          \u239f      \u2502 2\u2502         \n      a\u22c5\u0393(-1/4) \u22c5\u0393(5/4)                                   \u23a0      \u2502p \u2502         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  for \u2502\u2500\u2500\u2502 < 1, Max\n                                                                 \u2502 2\u2502         \n                                                                 \u2502a \u2502         \n                                                                              \n\u239e                                                                             \n\u239f\u22c5\u0393(1/4)\u22c5\u0393(3/4)                                                               \n\u239f                                                                             \n\u239f                                                                             \n\u239f                                                                \u2502 2\u2502         \n\u23a0                                                                \u2502a \u2502         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                              for \u2502\u2500\u2500\u2502 < 1     \n                                                                 \u2502 2\u2502         \n                                                                 \u2502p \u2502         \n                                                                              \n     \u2502  2\u239e                                                                    \n 1/4 \u2502 p \u239f                                                                    \n     \u2502 \u2500\u2500\u239f                                                                    \n4    \u2502  2\u239f                                                                    \n     \u2502 a \u23a0                                                                    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                                    otherwise       \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                          \u2502   \n(0, cos(Abs(periodic_argument(polar_lift(a)**2, oo))/2)*sqrt(Abs(a**2))), \u2502per\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                               \u239e\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                          2    \u239f\n              \u239b          2      \u239e\u2502       p     \u239f\niodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 < \u03c0 \u2227 \u2500\u2500 \u2260 1\u239f\n                                          2    \u239f\n                                         a     \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u239f\n                                               \u23a0"
      }
     ],
     "prompt_number": 176
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "_[0]",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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       "prompt_number": 177,
       "text": "\n\u23a7      \u239b                                                  \u239b       \u239b           \n\u23aa      \u239c                                                  \u239c   ___ \u239c         4\u22c5\n\u23aa      \u239c                                                  \u239c \u2572\u2571 2 \u22c5\u239c1 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa      \u239c                                                  \u239c       \u239c       \u239b   \n\u23aa      \u239c                                                  \u239c       \u239c     2 \u239c   \n\u23aa      \u239c                                                  \u239c       \u239c    a \u22c5\u239c-4 \n\u23aa      \u239c                                                  \u239c       \u239c       \u239c   \n\u23aa      \u239c                                            ___   \u239c       \u239d       \u239d   \n\u23aa      \u239c                                        2\u22c5\u2572\u2571 \u03c0 \u22c5p\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa      \u239c                                                  \u239c       ____________\n\u23aa      \u239c  ___ \u239b      1               1      \u239e             \u239c      \u2571       _____\n\u23aa      \u239c\u2572\u2571 \u03c0 \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f             \u239c     \u2571       \u2571     \n\u23aa      \u239c      \u239c      _______         _______\u239f             \u239c    \u2571       \u2571      \n\u23aa      \u239c      \u239c     \u2571     p         \u2571     p \u239f             \u239c   \u2571       \u2571   1 - \n\u23aa      \u239c      \u239c2\u22c5  \u2571  1 + \u2500    2\u22c5  \u2571  1 - \u2500 \u239f             \u239c  \u2571       \u2571        \n\u23aa  ___ \u239c      \u239d  \u2572\u2571       a      \u2572\u2571       a \u23a0             \u239d\u2572\u2571      \u2572\u2571         \n\u23aa\u2572\u2571 \u03c0 \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa      \u239c            \u0393(1/4)\u22c5\u0393(3/4)                                             \n\u23aa      \u239d                                                                      \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23a8                                                                   ___       \n\u23aa                                                                 \u2572\u2571 a        \n\u23aa                                                                             \n\u23aa                                            \u239b      1                1       \u239e\n\u23aa                                            \u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n\u23aa                                            \u239c      _______         _________\u239f\n\u23aa                                            \u239c     \u2571 a             \u2571   a     \u239f\n\u23aa                                            \u239c2\u22c5  \u2571  \u2500 + 1    2\u22c5  \u2571  - \u2500 + 1 \u239f\n\u23aa                                            \u239d  \u2572\u2571   p          \u2572\u2571     p     \u23a0\n\u23aa                                            \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                                                                    ___      \n\u23aa                                                                \u03c0\u22c5\u2572\u2571 p       \n\u23aa                                                                             \n\u23aa                                                               \u239b             \n\u23aa                                                   ___ \u256d\u2500\u256e2, 1 \u239c 3/4    1/4, \n\u23aa                                                 \u2572\u2571 \u03c0 \u22c5\u2502\u2576\u2510     \u239c             \n\u23aa                                                       \u2570\u2500\u256f3, 3 \u239c0, 1/2    1/4\n\u23aa                                                               \u239d             \n\u23aa                                                 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                                                                   ___       \n\u23a9                                                                 \u2572\u2571 a        \n\n 2     \u239e                                         \u239e       \u239e              \np      \u239f                                         \u239f       \u239f              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                                         \u239f       \u239f              \n     2\u239e\u239f                                         \u239f       \u239f              \n  4\u22c5p \u239f\u239f                                         \u239f       \u239f              \n+ \u2500\u2500\u2500\u2500\u239f\u239f                                         \u239f       \u239f              \n    2 \u239f\u239f                     ___  2              \u239f       \u239f              \n   a  \u23a0\u23a0                 4\u22c5\u2572\u2571 2 \u22c5p               \u239f       \u239f              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u0393(3/4)\u239f              \n________                                      3/2\u239f       \u239f              \n___                       \u239b      ________    \u239e   \u239f       \u239f              \n 2            \u239b        2\u239e \u239c     \u2571      2     \u239f   \u239f       \u239f              \np           2 \u239c     4\u22c5p \u239f \u239c    \u2571      p      \u239f   \u239f       \u239f              \n\u2500\u2500  + 1    a \u22c5\u239c-4 + \u2500\u2500\u2500\u2500\u239f\u22c5\u239c   \u2571   1 - \u2500\u2500  + 1\u239f   \u239f       \u239f              \n 2            \u239c       2 \u239f \u239c  \u2571         2     \u239f   \u239f       \u239f              \na             \u239d      a  \u23a0 \u239d\u2572\u2571         a      \u23a0   \u23a0       \u239f              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f              \n              2                                          \u239f      \u2502 2\u2502    \n     a\u22c5\u0393(-1/4) \u22c5\u0393(5/4)                                   \u23a0      \u2502p \u2502    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  for \u2502\u2500\u2500\u2502 < 1\n                                                                \u2502 2\u2502    \n                                                                \u2502a \u2502    \n                                                                        \n                                                                        \n\u22c5\u0393(1/4)\u22c5\u0393(3/4)                                                          \n                                                                        \n                                                                        \n                                                                \u2502 2\u2502    \n                                                                \u2502a \u2502    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                              for \u2502\u2500\u2500\u2502 < 1\n                                                                \u2502 2\u2502    \n                                                                \u2502p \u2502    \n                                                                        \n    \u2502  2\u239e                                                               \n1/4 \u2502 p \u239f                                                               \n    \u2502 \u2500\u2500\u239f                                                               \n    \u2502  2\u239f                                                               \n    \u2502 a \u23a0                                                               \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                                    otherwise  \n                                                                        \n                                                                        "
      }
     ],
     "prompt_number": 177
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt((sqrt(p**2-a**2)+p)/(p**2-a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAI0AAAA3BAMAAADDIdSFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 178,
       "text": "\n       ____________________\n      \u2571        ___________ \n     \u2571        \u2571    2    2  \n    \u2571   p + \u2572\u2571  - a  + p   \n   \u2571    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \n  \u2571            2    2      \n\u2572\u2571          - a  + p       "
      }
     ],
     "prompt_number": 178
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAR8AAAA4BAMAAAA2p+FEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3n8swhRDpBxFdAAAAABJRU5ErkJggg==\n",
       "prompt_number": 179,
       "text": "\n                       \u239b      ____________________________            \u239e\n                       \u239c     \u2571     p             1                    \u239f\nInverseLaplaceTransform\u239c    \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 , p, t, None\u239f\n                       \u239c   \u2571      2    2      ___________             \u239f\n                       \u239c  \u2571    - a  + p      \u2571    2    2              \u239f\n                       \u239d\u2572\u2571                 \u2572\u2571  - a  + p               \u23a0"
      }
     ],
     "prompt_number": 179
    },
    {
     "cell_type": "markdown",
     "source": "43"
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "tau = Symbol(\"tau\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 180
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2/sqrt(a*pi)*integrate(exp(-tau**2), (tau, 0, sqrt(a*t)))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAuBAMAAAB9m5IOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
       "prompt_number": 181,
       "text": "\n   \u239b  _____\u239e\nerf\u239d\u2572\u2571 a\u22c5t \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     ___    \n   \u2572\u2571 a     "
      }
     ],
     "prompt_number": 181
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 182,
       "text": "\n\u239b         1                                          \u03c0\u239e\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument(a, \u221e)\u2502 \u2264 \u2500\u239f\n\u239c            _______                                 2\u239f\n\u239c  ___      \u2571 a + p                                   \u239f\n\u239c\u2572\u2571 a \u22c5p\u22c5  \u2571  \u2500\u2500\u2500\u2500\u2500                                   \u239f\n\u239d        \u2572\u2571     a                                     \u23a0"
      }
     ],
     "prompt_number": 182
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAABEBAMAAABdQOseAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 183,
       "text": "\n                       \u239b      1                  \u239e\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239c      _______            \u239f\n                       \u239c     \u2571     p             \u239f\n                       \u239cp\u22c5  \u2571  1 + \u2500             \u239f\n                       \u239d  \u2572\u2571       a             \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                        ___                       \n                      \u2572\u2571 a                        "
      }
     ],
     "prompt_number": 183
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/(p*sqrt(p+a))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAqBAMAAAAaH/vSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAABP0lEQVQ4EWNgQANCBmgCGFzX/AIMMXQB/lEl6EEC5hMOl9b5qwnGAFajh5sg/38C\n4MPA+Ngfm7NQnTIRlYuFx7IBiyCqEGcDnM+CIzc5wVUwsB9AsJFZGggOqpKyjWc2guU4FBgY0o4v\nALOhSiByjAe/MewGCzM5MHAmcCMrgcoxi3xgiAIraWJgOMXADfEVxBSYHNsEhpkMQGmGLQwMVxm4\nQF7ZOXPeypmTgAywHAMD5wGGq8zAUoVxAgPHZwZ5AbCJULeA5IB8/gbmiwycDxhYHRhYPjKcYkdW\nApZjYPB3ABrO/IVBEih3g+MRG7ISsBwDw8kzIUDRlQzBQNL6jFAGshKIHIMaWKy+cQKYhhLMIFcz\nQOUYLoM5vNsMwDQqAZHj+AgWZfuJKgnmQeXsfoO9yfgdixKoHEzGAcbAoAHQC4P6oFf+PwAAAABJ\nRU5ErkJggg==\n",
       "prompt_number": 184,
       "text": "\n     1     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    _______\np\u22c5\u2572\u2571 a + p "
      }
     ],
     "prompt_number": 184
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAL8AAAAuBAMAAACL/tXTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nZOxwaHMRAfIAAAAASUVORK5CYII=\n",
       "prompt_number": 185,
       "text": "\n                \u239b  ___   ___\u239e\nHeaviside(t)\u22c5erf\u239d\u2572\u2571 a \u22c5\u2572\u2571 t \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n              ___            \n            \u2572\u2571 a             "
      }
     ],
     "prompt_number": 185
    },
    {
     "cell_type": "markdown",
     "source": "44"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "integrate(exp(-tau**2), (tau, 0, sqrt((b-1)*t)))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAlBAMAAAA6go8yAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nIHb5uN8AAAAASUVORK5CYII=\n",
       "prompt_number": 186,
       "text": "\n  ___    \u239b  ___________\u239e\n\u2572\u2571 \u03c0 \u22c5erf\u239d\u2572\u2571 t\u22c5(b - 1) \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n           2            "
      }
     ],
     "prompt_number": 186
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "_*2*exp(-a*t)/sqrt(pi*(b-a))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKQAAAA6BAMAAADM9+9BAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nsZIUi67ri17BF/wPB8pT4m9nBVgAAAAASUVORK5CYII=\n",
       "prompt_number": 187,
       "text": "\n -a\u22c5t    \u239b  ___________\u239e\n\u212f    \u22c5erf\u239d\u2572\u2571 t\u22c5(b - 1) \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         ________       \n       \u2572\u2571 -a + b        "
      }
     ],
     "prompt_number": 187
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMgAAAAuBAMAAACfa0syAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 188,
       "text": "\n                \u239b -a\u22c5t    \u239b  _________\u239e      \u239e\nLaplaceTransform\u239d\u212f    \u22c5erf\u239d\u2572\u2571 b\u22c5t - t \u23a0, t, p\u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                    ________                  \n                  \u2572\u2571 -a + b                   "
      }
     ],
     "prompt_number": 188
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/((p+a)*sqrt(p+b))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAAqBAMAAACdLhLUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 189,
       "text": "\n        1        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          _______\n(a + p)\u22c5\u2572\u2571 b + p "
      }
     ],
     "prompt_number": 189
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOYAAAAsBAMAAACDN9laAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nohFj/a76AAAAAElFTkSuQmCC\n",
       "prompt_number": 190,
       "text": "\n                       \u239b            1                        \u239e\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239c    _______       _______            \u239f\n                       \u239da\u22c5\u2572\u2571 b + p  + p\u22c5\u2572\u2571 b + p             \u23a0"
      }
     ],
     "prompt_number": 190
    },
    {
     "cell_type": "markdown",
     "source": "45"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Integral(exp(-tau**2), (tau, 0, sqrt(a*t))).doit()\n# Laplace transform should try the doit() on integrals.",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHQAAAAlBAMAAABhZtBGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nMsUE3p1ABG4x4W1t/TKFwfBf1YSoR+Q/TmYAAAAASUVORK5CYII=\n",
       "prompt_number": 191,
       "text": "\n  ___    \u239b  _____\u239e\n\u2572\u2571 \u03c0 \u22c5erf\u239d\u2572\u2571 a\u22c5t \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        2         "
      }
     ],
     "prompt_number": 191
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-a*t)/sqrt(pi*t)+2*sqrt(a/pi)*_",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALIAAAAvBAMAAAC1XI3GAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n95P4hi3UAAAAAElFTkSuQmCC\n",
       "prompt_number": 192,
       "text": "\n                         -a\u22c5t   \n  ___    \u239b  _____\u239e      \u212f       \n\u2572\u2571 a \u22c5erf\u239d\u2572\u2571 a\u22c5t \u23a0 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                       ___   ___\n                     \u2572\u2571 \u03c0 \u22c5\u2572\u2571 t "
      }
     ],
     "prompt_number": 192
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 193,
       "text": "\n\u239b          _______                                                            \n\u239c  ___    \u2571 a + p                                                             \n\u239c\u2572\u2571 a \u22c5  \u2571  \u2500\u2500\u2500\u2500\u2500                                                             \n\u239c      \u2572\u2571     a        \u239b                            \u03c0                         \n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u239c\u2502periodic_argument(a, \u221e)\u2502 < \u2500 \u2228 \u2502periodic_argument(a, \n\u239d        p             \u239d                            2                         \n\n                                                                              \n                                                                              \n                                                                              \n      \u03c0                               \u03c0\u239e   \u239b                            \u03c0     \n\u221e)\u2502 \u2264 \u2500 \u2228 \u2502periodic_argument(p, \u221e)\u2502 = \u2500\u239f \u2227 \u239c\u2502periodic_argument(a, \u221e)\u2502 < \u2500 \u2228 \u2502p\n      2                               2\u23a0   \u239d                            2     \n\n                                                                              \n                                                                              \n                                                                              \n                          \u03c0\u239e   \u239b                            \u03c0                 \neriodic_argument(a, \u221e)\u2502 \u2264 \u2500\u239f \u2227 \u239c\u2502periodic_argument(a, \u221e)\u2502 < \u2500 \u2228 \u2502periodic_argu\n                          2\u23a0   \u239d                            2                 \n\n                                                                              \n                                                                              \n                                                                              \n              \u03c0                               \u03c0\u239e   \u239b                          \nment(a, \u221e)\u2502 \u2264 \u2500 \u2228 \u2502periodic_argument(a, \u221e)\u2502 = \u2500\u239f \u2227 \u239c\u2502periodic_argument(a, \u221e)\u2502 \n              2                               2\u23a0   \u239d                          \n\n                                             \u239e\n                                             \u239f\n                                             \u239f\n  \u03c0                               \u03c0   -p    \u239e\u239f\n< \u2500 \u2228 \u2502periodic_argument(a, \u221e)\u2502 \u2264 \u2500 \u2228 \u2500\u2500 \u2260 1\u239f\u239f\n  2                               2   a     \u23a0\u23a0"
      }
     ],
     "prompt_number": 193
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "_[0]",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAA4BAMAAABDI8iLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMrsQmSKrRHZUzYnd\n72ZiHOLWAAACFElEQVRIDeVUPUscQRh+9svd9e70qhRJ4WKwUdBFgqV3IY1pRBBsd0FjZXFEsRAj\ny2HgQgr3BwSzVdoYCKRIc8RC0gliZWNhoShyxoiEFOZdb3f2I3OMhZ0Dxzxf8+7czMsAvOHcZEaL\nlwG+8OWcGuQ4l2pVrpwTC7WcwKXdkapz3ViciYAaC9z5XadUX3hA8blYgNEYAKiWUt98nlR6mkAo\nLjCh0BJKmUX/MbOkKoNAjw/tUgqezS/Mz6GEVWbJUwwCsg2zaWzd1sJrfGXWI4YIyIDuFtfaqe/K\nNvNeMURgk36Txy/bqaEj9hnNBRonTXLDMduegAKB3ZgAXWXIntqMBCsxIAUJocM+g1ql4yFN9xID\nozttEl4HHdwBij6wQkSljfSmu7BFKzd8KBa0a1RsaCM2EN91qiL97wCFMqRLnJkwHar3JGNHRL/A\nKMF97ZMBlNxO7fwD3yi1dD72lva0CwTE/h8fpy0mUuFUOzfqU8wpTdQYxl8YjKleJTGMXwnGYHiL\n0XDspGmg/IllmjfsuJ2BD3ifcsop7PhxOwPL2DlMWSlYdC3G1PN1n5EMUPfcDOcT/XeH5dn4jZ3l\nfNbPl3PqSY7fP003ZSfcuv/PPrCK5vr4kbgtxite+KwIxmIfCk1Bhuw6ZEucGkZpS5y6whvx7qUL\n/BSX6hk+LYtTXZ44Q+9c7S4pR7x1eow/v+DU+gd0jpvdUR6+OAAAAABJRU5ErkJggg==\n",
       "prompt_number": 194,
       "text": "\n          _______\n  ___    \u2571 a + p \n\u2572\u2571 a \u22c5  \u2571  \u2500\u2500\u2500\u2500\u2500 \n      \u2572\u2571     a   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        p        "
      }
     ],
     "prompt_number": 194
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAABEBAMAAAAM1Nh9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ncJjA/g/rKSt8wq7UYgAAAABJRU5ErkJggg==\n",
       "prompt_number": 195,
       "text": "\n                             \u239b    _______            \u239e\n                             \u239c   \u2571     p             \u239f\n                             \u239c  \u2571  1 + \u2500             \u239f\n  ___                        \u239c\u2572\u2571       a             \u239f\n\u2572\u2571 a \u22c5InverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                             \u239d     p                 \u23a0"
      }
     ],
     "prompt_number": 195
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sqrt(p+a)/p",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAqBAMAAAD7dO6DAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMrsQmSKrRHZUzYnd\n72ZiHOLWAAABXElEQVQ4EWNggADB/1jAB6gkgzKMgY1mdICLMk6AM2EMpgAYi4HzAZwJY4jBGAzY\nZFMZGDj65oCVQPXOcH7nDNXCmsDAoMHggyTL/PgHgwtUlk2Age8AlAfRy274gWEPVDaMgYF3A8NJ\nIM9JSeuQkhqQwaHAoASkgDIMjgwM8QXMn4EsuKuYHjDcZmBg75/AwKzAwCBvwHcA7FOoq/gD2C8D\n1TItYOAWAJocEJXQAOTCfCQvwANSzP6RwQhE+T3OAAcJVO+jdxtBahkOMWwH01CCHWw+QwaUOz9Y\nAVkWyr4OpXk9CjBlWSEeAPrsE6YkQ9UXA4go808ssgghAQSTmiwsiQ0u9IGaFg05szhb7J5AowaL\n2+3sG1gcsIhDhIrlGbgP4JRlaGJgUsAtqwfKEjjBV4Za3K5i/AjOSzg08+m9FsAhBRRma8AtB8xL\nBfhk43E7CZjR1sNKCpgRACjjcKdsFrtXAAAAAElFTkSuQmCC\n",
       "prompt_number": 196,
       "text": "\n  _______\n\u2572\u2571 a + p \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    p    "
      }
     ],
     "prompt_number": 196
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIkAAAAsBAMAAACj24anAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nyfwYsCoRTjr+f/B/AO/8DXDf/d1PAAAAAElFTkSuQmCC\n",
       "prompt_number": 197,
       "text": "\n                       \u239b  _______            \u239e\n                       \u239c\u2572\u2571 a + p             \u239f\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239d    p                \u23a0"
      }
     ],
     "prompt_number": 197
    },
    {
     "cell_type": "markdown",
     "source": "46"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "besselj(0, a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADIAAAASBAMAAADrvZC0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZonvmSLNu3ZEMt2r\nEFSC55RrAAABJElEQVQYGU2QP0jDQBTGf9GkjSn9YxwcCiJ0FKEIjoVOzsFJnDKqqFQnxyxCcRJx\nN4O7N+igKLgoOggZCwpmcHCsoEMHwXe5kHrw3vu+33cHdwf/llXosTLooUiqXS2t1neQISvNBqzD\nZyZLPUMqecA+lBJt6pFBZp/oEbgNzc5i3eHZDLwheKE2rZwIoD+nplYuI9CGW92gKqYc2UqauI4m\nX6z6AThtaGJ3qcdCr6RqP+wQSqJgiUrKhUBupGzlKTblbBtnxHvCrE46zAST81bInpgh3i/NGk+k\n2jyykLghB5JswbKz7bJrJXg91vxF5MyGJH04nX475ONIvqwhQJ6luJbxmhnTzO/AHXIvJo7H0Uku\nz/1Yq5fcyj3vC5mJQWHdhD/Btj3/r/FHfAAAAABJRU5ErkJggg==\n",
       "prompt_number": 198,
       "text": "besselj(0, a\u22c5t)"
      }
     ],
     "prompt_number": 198
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 199,
       "text": "\n\u239b       1             \u2502                 \u239b          2      \u239e\u2502    \u239e\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 = 0\u239f\n\u239c        ________                                               \u239f\n\u239c       \u2571      2                                                \u239f\n\u239c      \u2571      p                                                 \u239f\n\u239ca\u22c5   \u2571   1 + \u2500\u2500                                                \u239f\n\u239c    \u2571         2                                                \u239f\n\u239d  \u2572\u2571         a                                                 \u23a0"
      }
     ],
     "prompt_number": 199
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simplify(_[0])",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAA4BAMAAACh/9PyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nscFox6KaY22DVVfxDTW2rjVl94L9AAAAAElFTkSuQmCC\n",
       "prompt_number": 200,
       "text": "\n       1        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        ________\n       \u2571      2 \n      \u2571      p  \na\u22c5   \u2571   1 + \u2500\u2500 \n    \u2571         2 \n  \u2572\u2571         a  "
      }
     ],
     "prompt_number": 200
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJcAAAASBAMAAACkzFEhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 201,
       "text": "Heaviside(t)\u22c5besselj(0, a\u22c5t)"
      }
     ],
     "prompt_number": 201
    },
    {
     "cell_type": "markdown",
     "source": "47"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "besseli(0, a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAASBAMAAAAAiiu3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIonvq3bduxBUMplm\nzUT5W2AUAAABIElEQVQYGU2Qv0vDUBDHP6FpftSkDQguIo24iYh/Qhf3jA4VsomTIrgHdHNoJ1cz\n1Uk61cWCGSvYVRyzuyg4iDh4l7xWb7jv977fu3fvPfgXxZL/sUqyo6Vzpcza+kxq5ckYVo5fSc6p\nUVKDbgyvygdmpO4ToVNCX52bXDPYZQUwFrxT/miEjmCzf83by0TmVDwyzkDwnHtN0MjA+8Ha2ZRi\nFYKUB/iWwonA/yDMQinWpDPmmNa7Ojms7DEiLOECuqX3RaCv0NOcHnPcolo6zoJ0aCd+fYNuwSGu\nvCnMaUQbvSQczuAW1k/OZCYUp5nQns72I3+Sw7OcKDHCKeWSqXIT1e/IqsxRYXchQzuueWu6rcTO\nJNVxuSA1erLMxAH8AphCN7rbZN3xAAAAAElFTkSuQmCC\n",
       "prompt_number": 202,
       "text": "besseli(0, a\u22c5t)"
      }
     ],
     "prompt_number": 202
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 203,
       "text": "\n\u239b\u23a7                                         \u2502 2\u2502                               \n\u239c\u23aa                                         \u2502p \u2502                               \n\u239c\u23aa                 0                   for \u2502\u2500\u2500\u2502 < 1, Max(0, cos(Abs(periodic_a\n\u239c\u23aa                                         \u2502 2\u2502                               \n\u239c\u23aa                                         \u2502a \u2502                               \n\u239c\u23aa                                                                            \n\u239c\u23aa                                         \u2502 2\u2502                               \n\u239c\u23aa                1                        \u2502a \u2502                               \n\u239c\u23aa        \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500           for \u2502\u2500\u2500\u2502 < 1                           \n\u239c\u23aa                __________               \u2502 2\u2502                               \n\u239c\u23aa               \u2571    2                    \u2502p \u2502                               \n\u239c\u23a8              \u2571    a                                                        \n\u239c\u23aa        p\u22c5   \u2571   - \u2500\u2500 + 1                                                   \n\u239c\u23aa            \u2571       2                                                       \n\u239c\u23aa          \u2572\u2571       p                                                        \n\u239c\u23aa                                                                            \n\u239c\u23aa              \u239b               \u2502  2\u239e                                         \n\u239c\u23aa  ___ \u256d\u2500\u256e2, 1 \u239c 1/2    1/2, 0 \u2502 p \u239f                                         \n\u239c\u23aa\u2572\u2571 \u03c0 \u22c5\u2502\u2576\u2510     \u239c               \u2502 \u2500\u2500\u239f                                         \n\u239c\u23aa      \u2570\u2500\u256f3, 3 \u239c0, 1/2    0    \u2502  2\u239f                                         \n\u239c\u23aa              \u239d               \u2502 a \u23a0                                         \n\u239c\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   otherwise                             \n\u239d\u23a9                 a                                                          \n\n                                                                              \n                                                    \u2502                 \u239b       \nrgument(polar_lift(a)**2, oo))/2)*sqrt(Abs(a**2))), \u2502periodic_argument\u239dpolar_l\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                    2    \u239e\n   2      \u239e\u2502       p     \u239f\nift (a), \u221e\u23a0\u2502 < \u03c0 \u2227 \u2500\u2500 \u2260 1\u239f\n                    2    \u239f\n                   a     \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u239f\n                         \u23a0"
      }
     ],
     "prompt_number": 203
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/sqrt(p**2-a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAtBAMAAADcoWFIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 204,
       "text": "\n      1       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   ___________\n  \u2571    2    2 \n\u2572\u2571  - a  + p  "
      }
     ],
     "prompt_number": 204
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKIAAAASBAMAAAAqFhhdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 205,
       "text": "Heaviside(t)\u22c5besseli(0, -a\u22c5t)"
      }
     ],
     "prompt_number": 205
    },
    {
     "cell_type": "markdown",
     "source": "48"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-(a+b)/2*t)*besseli(0, (a-b)/2*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOQAAAAsBAMAAACHwglnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nMRP73XwXAAAAAElFTkSuQmCC\n",
       "prompt_number": 206,
       "text": "\n   \u239b  a   b\u239e                      \n t\u22c5\u239c- \u2500 - \u2500\u239f                      \n   \u239d  2   2\u23a0        \u239b     \u239ba   b\u239e\u239e\n\u212f           \u22c5besseli\u239c0, t\u22c5\u239c\u2500 - \u2500\u239f\u239f\n                    \u239d     \u239d2   2\u23a0\u23a0"
      }
     ],
     "prompt_number": 206
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQoAAAAsBAMAAABvZ+tlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nKgX3Mm79B/p/9iSB1IB1RQMAAAAASUVORK5CYII=\n",
       "prompt_number": 207,
       "text": "\n                \u239b   a\u22c5t   b\u22c5t                            \u239e\n                \u239c - \u2500\u2500\u2500 - \u2500\u2500\u2500                            \u239f\n                \u239c    2     2         \u239b   a\u22c5t   b\u22c5t\u239e      \u239f\nLaplaceTransform\u239c\u212f           \u22c5besseli\u239c0, \u2500\u2500\u2500 - \u2500\u2500\u2500\u239f, t, p\u239f\n                \u239d                    \u239d    2     2 \u23a0      \u23a0"
      }
     ],
     "prompt_number": 207
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/sqrt((p+a)*(p+b))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAtBAMAAACQZ671AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEWGmjtqKAAAAAElFTkSuQmCC\n",
       "prompt_number": 208,
       "text": "\n         1         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  _________________\n\u2572\u2571 (a + p)\u22c5(b + p) "
      }
     ],
     "prompt_number": 208
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAO4AAAA3BAMAAAD58Wv2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nb2UAAAAASUVORK5CYII=\n",
       "prompt_number": 209,
       "text": "\n                       \u239b            1                        \u239e\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239c   ______________________            \u239f\n                       \u239c  \u2571                    2             \u239f\n                       \u239d\u2572\u2571  a\u22c5b + a\u22c5p + b\u22c5p + p              \u23a0"
      }
     ],
     "prompt_number": 209
    },
    {
     "cell_type": "markdown",
     "source": "49"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-a*t)*besselj(0, sqrt(a**2-b**2)*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJkAAAAhBAMAAAA46xSRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
       "prompt_number": 210,
       "text": "\n             \u239b        _________\u239e\n -a\u22c5t        \u239c       \u2571  2    2 \u239f\n\u212f    \u22c5besselj\u239d0, t\u22c5\u2572\u2571  a  - b  \u23a0"
      }
     ],
     "prompt_number": 210
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANwAAAAhBAMAAABAeNh1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 211,
       "text": "\n                \u239b             \u239b        _________\u239e      \u239e\n                \u239c -a\u22c5t        \u239c       \u2571  2    2 \u239f      \u239f\nLaplaceTransform\u239d\u212f    \u22c5besselj\u239d0, t\u22c5\u2572\u2571  a  - b  \u23a0, t, p\u23a0"
      }
     ],
     "prompt_number": 211
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/sqrt(p**2+2*a*p+b**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAtBAMAAACe4hLfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 212,
       "text": "\n         1          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   _________________\n  \u2571          2    2 \n\u2572\u2571  2\u22c5a\u22c5p + b  + p  "
      }
     ],
     "prompt_number": 212
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAM4AAAA3BAMAAAC2rGgmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 213,
       "text": "\n                       \u239b         1                      \u239e\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239c   _________________            \u239f\n                       \u239c  \u2571          2    2             \u239f\n                       \u239d\u2572\u2571  2\u22c5a\u22c5p + b  + p              \u23a0"
      }
     ],
     "prompt_number": 213
    },
    {
     "cell_type": "markdown",
     "source": "50"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sinh(2*sqrt(t))/sqrt(pi)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAuBAMAAACmIDgsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nt2pk9Z5vxBFUo7FmaA6NalOTW4H/AZVqwSIjO+kdAAAAAElFTkSuQmCC\n",
       "prompt_number": 214,
       "text": "\n    \u239b    ___\u239e\nsinh\u239d2\u22c5\u2572\u2571 t \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n      ___    \n    \u2572\u2571 \u03c0     "
      }
     ],
     "prompt_number": 214
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 215,
       "text": "\n\u239b  1          \u239e\n\u239c  \u2500          \u239f\n\u239c  p          \u239f\n\u239c \u212f           \u239f\n\u239c\u2500\u2500\u2500\u2500, 0, True\u239f\n\u239c 3/2         \u239f\n\u239dp            \u23a0"
      }
     ],
     "prompt_number": 215
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALoAAAAuBAMAAABt1x6XAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 216,
       "text": "\n    \u239b    ___\u239e             \nsinh\u239d2\u22c5\u2572\u2571 t \u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n            ___           \n          \u2572\u2571 \u03c0            "
      }
     ],
     "prompt_number": 216
    },
    {
     "cell_type": "markdown",
     "source": "51"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sin(a*t)/t",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAAnBAMAAACoKJZjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZlTNRImrEN27MiJ2\n75lZwTvPAAABX0lEQVQ4Ea2SMUvDQBiG36TxorElVsGCKBRdxE0QdAyI4CQd/AH+ACGZ3LTBX+Ao\n4nAgCE4NuriUJpO/w82hiKIgVKF+d81dU+Gc8kLu3u97wpfL8QKTmtPl2KkWS5TDWe5WItWaz80s\nh52O/JqGcQ5nWsB27tWm3oYfAJnq5jujnlSH1hvAWl3wLvnud71PtU9PNVvHwQ5VVOyBRW3u/qBH\nH24TvMaLWIBahP3Agc/xLhZ0aVKMV2BAcDoBG16J/kCe4ZHeb+Ec3puAHNF9mGh4CvQC9wvWxWjs\nEqZSDekMnciKmyy15YEWUQ0IfsqxFY5a8nCUVpoN4A5YzjIv3Lr92Ag36TdSOM+N48Tuc6BOo4ty\n40L19/pwOIYOXe+kGN1ErhNl9O6m2j5pV7oZ/qPSP1buQMqyWSLLRoksGyWybJLMsgnKLBuhyLJJ\nMssmKLNsgjLLJiizbIIyywX4Cy0DaRmSOuonAAAAAElFTkSuQmCC\n",
       "prompt_number": 217,
       "text": "\nsin(a\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   t    "
      }
     ],
     "prompt_number": 217
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 218,
       "text": "\n\u239b    \u239ba\u239e     \u2502                 \u239b          2      \u239e\u2502    \u239e\n\u239catan\u239c\u2500\u239f, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 = 0\u239f\n\u239d    \u239dp\u23a0                                               \u23a0"
      }
     ],
     "prompt_number": 218
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJwAAAAsBAMAAABiXMwLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6eRMbVDTXxn1I2qLJ76kceBH1Es5J/HcB7VRYQAAAABJRU5ErkJggg==\n",
       "prompt_number": 219,
       "text": "\n                       \u239b    \u239ba\u239e            \u239e\nInverseLaplaceTransform\u239catan\u239c\u2500\u239f, p, t, None\u239f\n                       \u239d    \u239dp\u23a0            \u23a0"
      }
     ],
     "prompt_number": 219
    },
    {
     "cell_type": "markdown",
     "source": "52"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2/t*sin(a*t)*cos(b*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAAAnBAMAAAA7jOGEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZomZu1R2IjLvq90Q\nRM0XT9VLAAACrklEQVRIDbVVP2gTURj/XZrcXS6X5NKaOlQ0ODgpBpRMIuoguJ0UOnQ6/xAdKsRN\nEckJnTrYIogdHAKCIigNCNpFOBAE65BgQQQHsxREhRootkPT+H2Xe+/U6HKHv+G93/fnd++97x7v\nA/4CRfpCJl2AJbgkwhHMj6RtliWVxKjp665vvZI+Isp9YSl2wK4BU8IZzkVgnizNRcoJvUh9E0ZG\nkFNAzhOGnJvAXjKSJUBKZJCJXLMHqENn5EUfUFa+DZzj9CE8DTz6d0Dv/Bk2SHeXnHyG2yJYPNuZ\nXc6uXx1zyEMyVIvN7PFnZLExWbUwNW6BsmhdOnp31Fn9WvE5hYFsCU/MLbQcbRswSZJ0Rpo0UIiq\nqna0xUQZ+zkLaFGpe1jEaQqmPRoI2ZdoYwt5iwZoC8AERsrIuxR6QQrbPFavoX6Ts3AdMBp4zpnI\n1GhgLG24rHdBJdOawB1kbP+EWAY+ucA+2qzFWbhBy9rocnGQIZthatObUm8uQOuh5WEXh2j/Qj/D\nWXhDG6plN5UTFBT7V1wckHoqmb6NiRQewvbr17Kh1V3Ub3EW8lwCo2E4qbB+SgNXWN/m/eMzcE/7\nomJN8aDTMmoTuxMNHPGzkHORwUcvZ+8BZimboVTHrbmdD0trczsW/TtgtbByCZOXKUI2pqttjF2w\nOIu+5kCvHESq4gIFCo4epV3+hre/WMP312yGYbq/ak3vhg6fJdqh411IBTskCF8bpB08lo6A0C0J\nYDYEC2fDE3yFSKaEw9IRBF6LBKiepJKYjqAzA3JG2NFm80c0nVAly4JFm8/TPepHBegRM6OtG6je\nQ42j108WLsbRp/v9jTj6/6+lLhIL1EVigZ+UGPAfyuj6QReJrh90kRh6v4vE0M/H0LLU7yIxvuF3\nkRh67iJxwF3k3/gJLULbzyuaiT8AAAAASUVORK5CYII=\n",
       "prompt_number": 220,
       "text": "\n2\u22c5sin(a\u22c5t)\u22c5cos(b\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         t         "
      }
     ],
     "prompt_number": 220
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "#laplace_transform(_, t, p)\n# Assertion error",
     "language": "python",
     "outputs": [],
     "prompt_number": 221
    },
    {
     "cell_type": "markdown",
     "source": "53"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(exp(a*t)-1)/t*sin(b*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAqBAMAAABo0JlkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 222,
       "text": "\n\u239b a\u22c5t    \u239e         \n\u239d\u212f    - 1\u23a0\u22c5sin(b\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         t         "
      }
     ],
     "prompt_number": 222
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 223,
       "text": "\n\u239b\u23a7        \u239b    \u239b            \u239b    p\u239e\u239e    \u239e                                     \n\u239c\u23aa        \u239c    \u239ca\u22c5polar_lift\u239c1 - \u2500\u239f\u239f    \u239f                        \u2502          2\u2502\n\u239c\u23aa\u239b    p\u239e \u239c    \u239c            \u239d    a\u23a0\u239f    \u239f                        \u2502 2 \u239b    p\u239e \u2502\n\u239c\u23aa\u239c1 - \u2500\u239f\u22c5\u239catan\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f - \u03c0\u239f                        \u2502a \u22c5\u239c1 - \u2500\u239f \u2502\n\u239c\u23aa\u239d    a\u23a0 \u239d    \u239d         b         \u23a0    \u23a0       \u239bb\u239e   \u03c0          \u2502   \u239d    a\u23a0 \u2502\n\u239c\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - atan\u239c\u2500\u239f + \u2500  for 1 < \u2502\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2502\n\u239c\u23aa                     \u239b    p\u239e                  \u239dp\u23a0   2          \u2502      2    \u2502\n\u239c\u23aa           polar_lift\u239c1 - \u2500\u239f                                   \u2502     b     \u2502\n\u239c\u23aa                     \u239d    a\u23a0                                                \n\u239c\u23a8                                                                            \n\u239c\u23aa               \u239b            \u239b    p\u239e\u239e                                        \n\u239c\u23aa               \u239ca\u22c5polar_lift\u239c1 - \u2500\u239f\u239f                                        \n\u239c\u23aa   \u239b    p\u239e     \u239c            \u239d    a\u23a0\u239f                                        \n\u239c\u23aa   \u239c1 - \u2500\u239f\u22c5atan\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                                        \n\u239c\u23aa   \u239d    a\u23a0     \u239d         b         \u23a0       \u239bb\u239e   \u03c0                          \n\u239c\u23aa   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - atan\u239c\u2500\u239f + \u2500           otherwise      \n\u239c\u23aa                     \u239b    p\u239e               \u239dp\u23a0   2                          \n\u239c\u23aa           polar_lift\u239c1 - \u2500\u239f                                                \n\u239d\u23a9                     \u239d    a\u23a0                                                \n\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n     \u239b\u2502                 \u239b          2      \u239e\u2502       \u2502                 \u239b        \n, 0, \u239d\u2502periodic_argument\u239dpolar_lift (p), \u221e\u23a0\u2502 = \u03c0 \u2228 \u2502periodic_argument\u239dpolar_li\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n                   \u239b\u2502                 \u239b                                2      \n  2      \u239e\u2502    \u239e   \u239c\u2502                 \u239c 2\u22c5\u2148\u22c5\u03c0           2               \u239b    p\nft (b), \u221e\u23a0\u2502 = 0\u23a0 \u2227 \u239c\u2502periodic_argument\u239c\u212f     \u22c5polar_lift (a)\u22c5polar_lift \u239c1 - \u2500\n                   \u239d\u2502                 \u239d                                 \u239d    a\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n    \u239e\u2502       \u2502                 \u239b                                2          \u239e\u2502 \n\u239e   \u239f\u2502       \u2502                 \u239c 2\u22c5\u2148\u22c5\u03c0           2               \u239b    p\u239e   \u239f\u2502 \n\u239f, \u221e\u239f\u2502 < \u03c0 \u2228 \u2502periodic_argument\u239c\u212f     \u22c5polar_lift (a)\u22c5polar_lift \u239c1 - \u2500\u239f, \u221e\u239f\u2502 \n\u23a0   \u23a0\u2502       \u2502                 \u239d                                 \u239d    a\u23a0   \u23a0\u2502 \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n   \u239e   \u239b                                               2    \u239e   \u239b\u2502            \n   \u239f   \u239c\u2502                 \u239b          2      \u239e\u2502       -b     \u239f   \u239c\u2502            \n= \u03c0\u239f \u2227 \u239c\u2502periodic_argument\u239dpolar_lift (b), \u221e\u23a0\u2502 = 0 \u2228 \u2500\u2500\u2500 \u2260 1\u239f \u2227 \u239c\u2502periodic_arg\n   \u23a0   \u239c                                               2    \u239f   \u239d\u2502            \n       \u239d                                              p     \u23a0                 \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n     \u239b                                2          \u239e\u2502                           \n     \u239c 2\u22c5\u2148\u22c5\u03c0           2               \u239b    p\u239e   \u239f\u2502       \u2502                 \u239b \nument\u239c\u212f     \u22c5polar_lift (a)\u22c5polar_lift \u239c1 - \u2500\u239f, \u221e\u239f\u2502 = \u03c0 \u2228 \u2502periodic_argument\u239dp\n     \u239d                                 \u239d    a\u23a0   \u23a0\u2502                           \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n                      \u239e   \u239b\u2502                 \u239b                                \n         2      \u239e\u2502    \u239f   \u239c\u2502                 \u239c 2\u22c5\u2148\u22c5\u03c0           2              \nolar_lift (b), \u221e\u23a0\u2502 = 0\u239f \u2227 \u239c\u2502periodic_argument\u239c\u212f     \u22c5polar_lift (a)\u22c5polar_lift\n                      \u23a0   \u239c\u2502                 \u239d                                \n                          \u239c                                                   \n                          \u239c                                                   \n                          \u239d                                                   \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n2          \u239e\u2502             2        \u239e   \u239b      2                               \n \u239b    p\u239e   \u239f\u2502           -b         \u239f   \u239c    -b            \u2502                 \u239b \n \u239c1 - \u2500\u239f, \u221e\u239f\u2502 < \u03c0 \u2228 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u2260 1\u239f \u2227 \u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u2260 1 \u2228 \u2502periodic_argument\u239dp\n \u239d    a\u23a0   \u23a0\u2502                 2    \u239f   \u239c          2                           \n                     2 \u239b    p\u239e     \u239f   \u239c 2 \u239b    p\u239e                            \n                    a \u22c5\u239c1 - \u2500\u239f     \u239f   \u239ca \u22c5\u239c1 - \u2500\u239f                            \n                       \u239d    a\u23a0     \u23a0   \u239d   \u239d    a\u23a0                            \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n                      \u239e   \u239b\u2502                 \u239b                                \n         2      \u239e\u2502    \u239f   \u239c\u2502                 \u239c 2\u22c5\u2148\u22c5\u03c0           2              \nolar_lift (b), \u221e\u23a0\u2502 = 0\u239f \u2227 \u239c\u2502periodic_argument\u239c\u212f     \u22c5polar_lift (a)\u22c5polar_lift\n                      \u239f   \u239d\u2502                 \u239d                                \n                      \u239f                                                       \n                      \u239f                                                       \n                      \u23a0                                                       \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n2          \u239e\u2502                                                 \u239e               \n \u239b    p\u239e   \u239f\u2502       \u2502                 \u239b          2      \u239e\u2502    \u239f   \u2502           \n \u239c1 - \u2500\u239f, \u221e\u239f\u2502 < \u03c0 \u2228 \u2502periodic_argument\u239dpolar_lift (b), \u221e\u23a0\u2502 = 0\u239f \u2227 \u2502periodic_ar\n \u239d    a\u23a0   \u23a0\u2502                                                 \u23a0               \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                              \u239e\n                              \u239f\n                              \u239f\n                              \u239f\n      \u239b          2      \u239e\u2502    \u239f\ngument\u239dpolar_lift (b), \u221e\u23a0\u2502 = 0\u239f\n                              \u239f\n                              \u239f\n                              \u239f\n                              \u239f\n                              \u239f\n                              \u239f\n                              \u239f\n                              \u239f\n                              \u239f\n                              \u239f\n                              \u239f\n                              \u239f\n                              \u23a0"
      }
     ],
     "prompt_number": 223
    },
    {
     "cell_type": "markdown",
     "source": "54"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "-EulerGamma-log(t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAASBAMAAABWXMbBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIrt2MqvNVGaJmd3v\nRBDLtyOEAAABYUlEQVQoFXWSMUvDUBSFTxrTpI1NQ1cdiv6BiIMtgoSiXf0hgmQR10IHO0lARRCE\nQAYdHArOgj+hc6csLiJo8SFu9d60IblRD7z77ndy7uM9CPCPBuynhZs/pPnCNFzGV+FJ6CnBZwAd\nYXrClCAHxkAlBI5lRpAY4KObE6AjIhLEgEHhQ/relxlBCmaHDoy7Q7pKE5gexeleCD08wcpR4Qa1\n0IjqkQ+8k79JazWhkqlyt4v9DAClzaDNbM96Jm+dFt/RdqlkOkVjEKK1QeLDlPUBfOttg26ENWCF\nELaPPEH91oDMpdKBz8Zwh/kNsNq0iysRbydUllKagqZqC6JHG55ZfjTQzdK0K1yhHunnqVX1UQ2o\n7RUC3F7mfPAVWPE9nPn8kUzdgxn7wCgPpB2/T6rlOv0AcMYLu/xr/B44Aeo0gNt0oFEKOCWmkHkx\nuuaskXB94VKQOSmAbB2PeU+a0IEfWCJIrZcSO1wAAAAASUVORK5CYII=\n",
       "prompt_number": 224,
       "text": "-log(t) - EulerGamma"
      }
     ],
     "prompt_number": 224
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 225,
       "text": "\n\u239blog(p)         \u239e\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500, 0, True\u239f\n\u239d  p            \u23a0"
      }
     ],
     "prompt_number": 225
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANQAAAASBAMAAADRQe2CAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+BoTt1arviaJqB1KWwB7be3D4RPzF5pz4AReKYitAAAAAElFTkSuQmCC\n",
       "prompt_number": 226,
       "text": "(-log(t) - EulerGamma)\u22c5Heaviside(t)"
      }
     ],
     "prompt_number": 226
    },
    {
     "cell_type": "markdown",
     "source": "55"
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "k = Symbol(\"k\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 227
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "phi = Sum(1/k, (k, 1, n))- EulerGamma\nphi",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAA0BAMAAAD8jeImAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 228,
       "text": "\n                n    \n               ____  \n               \u2572     \n                \u2572   1\n                 \u2572  \u2500\n-EulerGamma +    \u2571  k\n                \u2571    \n               \u2571     \n               \u203e\u203e\u203e\u203e  \n              k = 1  "
      }
     ],
     "prompt_number": 228
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "t**n/factorial(n)*(phi-log(t))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANgAAAAqBAMAAAAjVIjMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 229,
       "text": "\n   \u239b                         n    \u239e\n   \u239c                        ____  \u239f\n   \u239c                        \u2572     \u239f\n   \u239c                         \u2572   1\u239f\n n \u239c                          \u2572  \u2500\u239f\nt \u22c5\u239c-log(t) - EulerGamma +    \u2571  k\u239f\n   \u239c                         \u2571    \u239f\n   \u239c                        \u2571     \u239f\n   \u239c                        \u203e\u203e\u203e\u203e  \u239f\n   \u239d                       k = 1  \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                 n!                "
      }
     ],
     "prompt_number": 229
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 230,
       "text": "\n\u239b  -n - 1                                                                     \n\u239c-p      \u22c5(-n\u22c5log(p) - n\u22c5harmonic(n) + n\u22c5polygamma(0, n) + EulerGamma\u22c5n + 1)  \n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, \n\u239d                                     n                                       \n\n                                                                              \n                                                \u239b                             \n0, (-n < 1 \u2228 1 < -n + 1) \u2227 0 < n + 1 \u2227 -n < 1 \u2227 \u239c-n < 1 \u2228 \u2502periodic_argument(p\n                                                \u239d                             \n\n                                                        \u239e\n        \u03c0\u239e   \u239b                                        \u03c0\u239e\u239f\n, \u221e)\u2502 = \u2500\u239f \u2227 \u239c0 < n + 1 \u2228 \u2502periodic_argument(p, \u221e)\u2502 = \u2500\u239f\u239f\n        2\u23a0   \u239d                                        2\u23a0\u23a0"
      }
     ],
     "prompt_number": 230
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simplify(_[0])",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgQAAAAoBAMAAABz3f/BAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n5A8H16ZYdJX/ANX46y0rCpx3AAAAAElFTkSuQmCC\n",
       "prompt_number": 231,
       "text": "\n -n - 1                                                                  \np      \u22c5(n\u22c5log(p) + n\u22c5harmonic(n) - n\u22c5polygamma(0, n) - EulerGamma\u22c5n - 1)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                    n                                    "
      }
     ],
     "prompt_number": 231
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "log(p)/p**(n+1)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGUAAAAUBAMAAABmLO6YAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 232,
       "text": "\n -n - 1       \np      \u22c5log(p)"
      }
     ],
     "prompt_number": 232
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYUAAAArBAMAAABlbIlRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 233,
       "text": "\n  n                                            \n-t \u22c5(log(t) - polygamma(0, n + 1))\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                    \u0393(n + 1)                   "
      }
     ],
     "prompt_number": 233
    },
    {
     "cell_type": "markdown",
     "source": "56"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(log(t)+EulerGamma)**2-pi**2/6",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJoAAAAlBAMAAABITe2EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
       "prompt_number": 234,
       "text": "\n                          2\n                     2   \u03c0 \n(log(t) + EulerGamma)  - \u2500\u2500\n                         6 "
      }
     ],
     "prompt_number": 234
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 235,
       "text": "\n\u239b   2            \u239e\n\u239clog (p)         \u239f\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, True\u239f\n\u239d   p            \u23a0"
      }
     ],
     "prompt_number": 235
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYcAAAAlBAMAAABbkzgcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/1zLv7C4qan0FjDPAAAAAElFTkSuQmCC\n",
       "prompt_number": 236,
       "text": "\n\u239b     2                              2               2\u239e             \n\u239d6\u22c5log (t) + 12\u22c5EulerGamma\u22c5log(t) - \u03c0  + 6\u22c5EulerGamma \u23a0\u22c5Heaviside(t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                 6                                  "
      }
     ],
     "prompt_number": 236
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "logcombine(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAnEAAAAlBAMAAADfI6LDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 237,
       "text": "\n                                                           2                  \n   2                                                      \u03c0 \u22c5Heaviside(t)     \nlog (t)\u22c5Heaviside(t) + 2\u22c5EulerGamma\u22c5log(t)\u22c5Heaviside(t) - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + Eu\n                                                                 6            \n\n                      \n        2             \nlerGamma \u22c5Heaviside(t)\n                      "
      }
     ],
     "prompt_number": 237
    },
    {
     "cell_type": "markdown",
     "source": "57"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/t*(exp(b*t)-exp(a*t))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAApBAMAAACGt2nyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIrt2ZpmJ70QQVDLN\n3atdH5w+AAABTklEQVQ4EWNgQAWcG1D5DAxchQwMs9EFGRhYHdDFOAsYGELQBRkYeAXQxbgeMDBc\nRBM0NpJIt0ATY2BViW4rbkARZTWYz7AORQTE4W1oAhmKDOQn7IG7ZhdcYjrDGnSHr2UwY7jNCVGx\nBK6wjyGMd8MEOBfE6GO44fChGyKEULiP8yY/B1KgsZgEsrqYLghZgKIQJGoqwOoFEQSTagzCSDwG\nqIlookAVbAUMiZgKkUUZlUDAgF/FBBHUrEpKZ5WUHjAwoIhCzJFvQDaPAWo1uihQzXwBBmjAQDRA\n3IguCpRjWsDggWwmRCG6KFAFZ4gzwo1APkQhuiiyUVB2FxaxkSj0nzjwYdiHDSNSnsPrWaYEvNII\nSf4GBBsvaz9eWYRk69lgBAcvSxmvLJLkbyQ2PiY3sQmARQGfMUhyfAZcSDw8TJ4FnnhkkaS4grFF\nIQAyPl81qGMXBgAAAABJRU5ErkJggg==\n",
       "prompt_number": 238,
       "text": "\n   a\u22c5t    b\u22c5t\n- \u212f    + \u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n      t      "
      }
     ],
     "prompt_number": 238
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANsAAAAuBAMAAABT8nHZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYOTyqIuNKmZg9Y9USNr/SMW+jeA/RTJfzTI+suUAAAAASUVORK5CYII=\n",
       "prompt_number": 239,
       "text": "\n                  \u239b a\u22c5t      \u239e                   \u239b b\u22c5t      \u239e\n                  \u239c\u212f         \u239f                   \u239c\u212f         \u239f\n- LaplaceTransform\u239c\u2500\u2500\u2500\u2500, t, p\u239f + LaplaceTransform\u239c\u2500\u2500\u2500\u2500, t, p\u239f\n                  \u239d t        \u23a0                   \u239d t        \u23a0"
      }
     ],
     "prompt_number": 239
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "log((p-a)/(p-b))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG0AAAAsBAMAAACd0iusAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nBcXbyHBzHkl/AAAAAElFTkSuQmCC\n",
       "prompt_number": 240,
       "text": "\n   \u239b-a + p\u239e\nlog\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n   \u239d-b + p\u23a0"
      }
     ],
     "prompt_number": 240
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAARkAAAAsBAMAAACj/tGOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
       "prompt_number": 241,
       "text": "\n                       \u239b   \u239b    a        p   \u239e            \u239e\nInverseLaplaceTransform\u239clog\u239c- \u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u239f, p, t, None\u239f\n                       \u239d   \u239d  -b + p   -b + p\u23a0            \u23a0"
      }
     ],
     "prompt_number": 241
    },
    {
     "cell_type": "markdown",
     "source": "58"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2/t*sinh(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE0AAAAnBAMAAAC4SNi/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 242,
       "text": "\n2\u22c5sinh(a\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     t     "
      }
     ],
     "prompt_number": 242
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJAAAAAsBAMAAAB4YCyFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 243,
       "text": "\n                  \u239bsinh(a\u22c5t)      \u239e\n2\u22c5LaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, t, p\u239f\n                  \u239d    t          \u23a0"
      }
     ],
     "prompt_number": 243
    },
    {
     "cell_type": "markdown",
     "source": "59"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2*(cos(b*t)-cos(a*t))/t",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAK8AAAAnBAMAAACLXwXzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIrt2ZomZVDLvq90Q\nRM3z6h40AAACsUlEQVRIDe2WPWgTYRjH/5e79PJxaeKkTob4vV01BUGUYCOKFQ0IHVwMxlkCFV2D\ng242IuIi2NLJohgHxcHhRHDQoQFpHRzsptglsQUztJzPex/vvZfctdo2S/EZ7n0+fs8/9z73Jjlg\nLVN40fN4qsfhDHd6EDcx5TrQytwNdaZwr2MVffDpq9WeDsVN7QLGfVVl7KAvZgHRsRatfljOxJZ7\n0KSb2Q8MGlZw104NYaJge96VaGWBQg9mtbiOp2z1Gb/LVUAuWKUbNvAAiboPpYDoRJlWD2ZEsoQr\nBnNEe+4EbIexBStwhF8huiKSzCc6TrMTYIc4TKtayUPN56HcukARm9jRM43U9de6HQCOMBCpB9B7\nipdE2BbWlmg9id21MURLc0gY0Eg4og806EKlxxbGhWcp10NfbMZrAmwLR8qA9hsf3rUhtb82U4BE\nNzWKgTLSNUJeWhgXPhBEzyCpu7CUY0YfP0yNKTqHSgtYTpiHSLgBPESyigmm+QYYyOUWc7lJFsml\nIPoR0gUXZhQzVdc8dMnY+SIDrQ5pFbMGTjHAP4pxGPw2OK218S0jwKwNnyFbm8OODqTOWURpEy3E\nVjCqYhrVroeX0qGzwflpeigzIsx0Y/uKR2g5DiVzmcZ6DnIT+AHcl37K+K4YiGUZ5p6K28URinvo\nRuqtCLOGuGn+okWdPgGlcg0jdOToqAHzxY9DuHmMRl+gBBd+YprZAPrOsCHCVkfQ5ZOQdL7SX4RU\nlyvQDtwFeGGUxuHanOuErgK9LvyMq2iT3A11OL0+/J6LyAZ3Qx1O/w0cqrINC2Z/rLUNR/V/S0ET\nkGpB2S3IRUpbIBIkkRZ+7ILqG85Zf5Mb7g5vnF+shBc3Vdm7qe41mumloy9mvfz2Q1nJ9kOVNBO6\n2h/lwer5/girldo/CP8BftoOfZ8mLvIAAAAASUVORK5CYII=\n",
       "prompt_number": 244,
       "text": "\n-2\u22c5cos(a\u22c5t) + 2\u22c5cos(b\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n           t            "
      }
     ],
     "prompt_number": 244
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATQAAAAsBAMAAAAZXVnuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIrt2ZomZVDLvq90Q\nRM3z6h40AAAFQklEQVRYCd2ZXWgcVRTH/5OdZDe7s8nigyiiu8SPog86mio+VFzbxPoQ7IIoioqh\nCQVpkIVIkAi6FF980ASK2odKV6KCUHFVrEUUBwStGtigVIRCDL4oKnTbBK2asp5zZvZj5s5u7mz6\n5IW9c+659/z2f8+9M7OXBTqUhfq60pNveti6pV5rtrc2uvKi4VbVb0tmmr4vxYoiLZSXOFsSUDTc\nd00ZTWOvZxklxG22o0gL5y0QJTIuBFXxpPUVgD8ugTTi3U+YyDhVmpspYg2vADu3L415b/eCU6Ul\nSZEUXoWX2apJU68K571GwZFxgpobzWN2Tx57n16jZBEmtvNZ/PD7lNuILi3IMzbG7eg4lhZbM470\nT+LBdAHHgSp5rsVBroBBh6qoWVN45kUciY5jadWiNZPNIHv1Z6DFvBIwKzgAXKCuVKYHaQovWcaH\n0XEs7acSsI8WL//eebKuolwV8DESnKwUOSJnTeUVsREd1y5tlzF/DvgayK5YmzBzpKqnBW1Ia/Kq\nmfS56Dh3QWFkS8i+UMIDsvMXHLNSTNrxHm+DajHAqyJZjo5jabEK7ukv44BZxl3AUAmDmWvK9lBx\nAjhE3ZEXVOGlcMaJjmNpmB9dwb135M1ReoAgZiM9PbGUiU+VgDHudu9Qk024tZihVSgvMfUYdHHx\ns2UPLCjfl1iVtmbbi+oNcVuTbb0t87l/PLs7rxvOfGgHM3a8uNkRhce9LrrQk46KZM0suu5Z99Ko\nx/ezv++ZY47rUaW18bribsVCHkgWBo66JISgkt630IhvZJRIS3kBQ25vwlUayyQ2qOMtPOm6u/M6\n44hxGMlF4HnHIFtKiDTLbnRil1girZGtWF58cfcyaOMdIL6JjHgRJq3F64yj6I/QT0v5m8ehS4i0\nVqdnibR3vUZiTQxPWqqAJxwMlb1OLV4YTuL7FmH81STpSyPg7MzYTON+9aQR6EZ6RBebQI2pirQA\nTuKrNmKUOMAqca2BktvAqsGaWOdX61EOQ1OatQ7c7bDrFFcaPJYWxHEobqCHao6NgTzXjKp3K9TP\nLGMR6VM1PAq8T602aX2TwM/imuZ6Kx4NCeIuG6FCP3NiBfqlYTMkleFaY5Yuq0KxOYwAJylq/8jN\nH4zcxPG4nVJAiaMimjV4NRpr+HASjlk4bv7Tj9BLSV+atUhPrzxeCS5o3La8DXL68Ioej6UFceRK\n27BxgoxxzFOtL41TN5xJ00ZjcmtBv0cMqUl6gRXiZfZrZi2Io8hDY3tyZv3fN1/6FLI1dFCumF/p\n130pVUQixwoat0HiurHbsK/+yX1nnNbelQGdK5lbAEejj9XrucuX5z5/1cFJi6M1ZulKGwV2jz1M\n6clzWEPaYL1+3pj56uDfM/Sr0yG/Bk+kBXCMbJXaj2yrqCtaQzxLWN8CS9wecsSbLsqlrRpOl6il\n8Az2+koYzjcA0wJXULjeP4xawupfwevcc1rp9hwDcupXeHwy9hc9XMgscdEPopawcNzgJ7VVVrp9\nDkUaH7T9RRenoOR4Ecb6YvVPh55t9OlWFN6CMtqVtjUuiEo/dcIOwlxW0BveDvLkZOwfqosLotCn\nKPMW1M/v1FJ4fND2l56lDZf8IGrpsjhQkXahZ5yCUvfGtqR13LqKYsWhSNutDNmWNDOn8HQXQZG2\nhGIQpsviuCBPTsZ+oC4uiMIvpuMnbW+vycnYD+xZ2tydfhC1dFkcGJyqnIz9RF1cEOWnuC1dFo/W\n4OniNFDbylrIVP8H0laXnZCJtbusZd1pctSWPH1c2B837cIuxR9BPp7yv9J/F0MNhEDeTmIAAAAA\nSUVORK5CYII=\n",
       "prompt_number": 245,
       "text": "\n                    \u239bcos(a\u22c5t)      \u239e                     \u239bcos(b\u22c5t)      \u239e\n- 2\u22c5LaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, t, p\u239f + 2\u22c5LaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, t, p\u239f\n                    \u239d   t          \u23a0                     \u239d   t          \u23a0"
      }
     ],
     "prompt_number": 245
    },
    {
     "cell_type": "markdown",
     "source": "60"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2*(cosh(b*t)-cosh(a*t))/t",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMMAAAAnBAMAAABAhOENAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnT7uAAAAAElFTkSuQmCC\n",
       "prompt_number": 246,
       "text": "\n-2\u22c5cosh(a\u22c5t) + 2\u22c5cosh(b\u22c5t)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n            t             "
      }
     ],
     "prompt_number": 246
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAUgAAAAsBAMAAAD1KDz4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIrt2ZomZVDLvq90Q\nRM3z6h40AAAFPklEQVRYCd2ZX2gcRRzHv5vb5C53e3/wQRDRO6JY9EFXU31q8TQX60OwB6IoKoYm\nFKR5OIiUEkGP4IuCJlDUPlR6UhWESk9FLaK4IGjVwB1KRSjEID6ICr22oVZNPX+/mb1/u7OX2Vye\nHMjsb38z89nP7uzu3VyAgLLUvKhuybfTHN3WbLT3Nw82h4ZjrnoOaZRlIp5pN3wpojCSwdDYOckP\nx/yu7SKDxctyu8fNk3TU5jiMZB/oEqFCM708uJJVV3KoCPwxqGQX9AFihWYGSMqrR8B0Hdi5TZIM\nfXsrzADJOLmJwtPzMkcNsatX9YG+RoTQTME7OJ7H/EQee55ew+XCDHHS9BfZ+Qx++J32eCe8pBJq\nrE/a4ZksGVkzjgxP46FkESdwCVkHqFH6BhzgChilRGhJNdS8giPhmSxZK1lz2Qyy132GOtaRLgPX\nAGYV+yFu+URmC5JqaLyCD8IzWfInstpLs5o/eaFMBH5WrqXrV8THiPGtmKD20FcyAFrCenhmt+Qu\nY+G8K/k1kK1bGzBz5Lel6W5J9kJrmeT58Ew53TCyZWSfL+NBVzJNz6BjVktxO7rFB6dWUkFriFfC\nM1kyUsU9wxXsNyvYjStiulNljGaur9ip0hSwSH1CT7camsBZJzyTJbEwXsd9d+XN8Yn8i/8eOvkL\nedtIzk4dz0RnykCB+8j3pMkhZC1CZRUMjc08Dl1m9FzFpQue/0hWtSvX9bH4hkhb012tnfDZv91Y\nA9qPaT68g0E7Xtjoz8MTbjtt6PVJRVxJsyTT83LTqif3cX7o0DFHZgIku6B9mbdjKQ/EiyNHJQ5B\nvLh7POr2jegqJBPuqJRsjUnnSCa2Tg1v4SmZ1oAGMwl0GPFl4DnHoFiUIEnLbvXALhEJydYVjORF\nLio3ozbeAaIbyIgsAiU70GAmIT7EME30by6MNkGSnR5uJCTfdXdiayJwJRNFPOkgVXEb9aEqpoAM\nLcP4s40LKUnU+bnCXOtZdyWJdjO9/Ettqu6ZC0kPU0BqNiJ0MQGrzLUuTzw4VgPW1EX+SD/KY9GW\ntGihdLfDqdNc6UJZ0svk8bgJSOU4GMlzzbzmpoU6MdBYRvJ0A48B79Fel+TQNPCzSM1yrQWlfl7m\nVWNU6HtXpEjfemwmJTJc6560BFYJkMMYcIqG7hu79f2xWxiCO+mK0MWkIux1oQ0aYPQwBQPzcOTE\nJB/dzRldnpzuZXob5vGKd7qjtuXeQ2cO10NAWdLyMCmVtGHjIwomsUB1SEk2TWeSdEMyvjPd3yOC\nxDR9XBajFc7rnrmgeJg0fLEwkTOb/7z50qcQd482T2r9Sl+OyokSYjl2aT04sRsLd2Bv85P7zzqd\nG1102KQSkh4mDTnWbOauXjn4+asOTlmM0D1pKTkO3Ft4hC5Znse2JEebzQvG3FcH/pqj78cO5XWh\nQtLDZG6nNH7kWMUzyp1e7UgAvwWOcybliHyyJDZdVTpZpj1dqIrZBaNwVhxBxePFu68I4HAdr3PL\nGV+zmxgRv3zoQvWY6pPmRY6vCCBOGPxBYFV8zT0JlaQKqstU8ZZ6jujuSOAXq5ccelfSX7+iC9Vl\nKnhi8e5zkEBfWpnQheoyFTyxePcdXBfIA3WhukwVz/1hrddTFxgkqYLqMhWSYvHea0h7usAASSVU\nl6mQNHM+w8ElldABJMXi3eepC+SBijNXQnWZCp5YvG+3pBI6gKRYvG+3pBI6gKTPTyR0gdxZMT1K\nqC5Tlzfo0/3/l1xdcZQn6UlaK7pTwwP1oPrMwP+w9Vpu07/teqA+5n+UOEEWgHJdUwAAAABJRU5E\nrkJggg==\n",
       "prompt_number": 247,
       "text": "\n                    \u239bcosh(a\u22c5t)      \u239e                     \u239bcosh(b\u22c5t)      \u239e\n- 2\u22c5LaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, t, p\u239f + 2\u22c5LaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, t, p\u239f\n                    \u239d    t          \u23a0                     \u239d    t          \u23a0"
      }
     ],
     "prompt_number": 247
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "log((p**2-a**2)/(p**2-b**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAAtBAMAAABxIHnhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nCynLz2IAAAAASUVORK5CYII=\n",
       "prompt_number": 248,
       "text": "\n   \u239b   2    2\u239e\n   \u239c- a  + p \u239f\nlog\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n   \u239c   2    2\u239f\n   \u239d- b  + p \u23a0"
      }
     ],
     "prompt_number": 248
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATkAAAAtBAMAAAAn/wH7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
       "prompt_number": 249,
       "text": "\n                       \u239b   \u239b       2           2   \u239e            \u239e\n                       \u239c   \u239c      a           p    \u239f            \u239f\nInverseLaplaceTransform\u239clog\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f, p, t, None\u239f\n                       \u239c   \u239c     2    2      2    2\u239f            \u239f\n                       \u239d   \u239d  - b  + p    - b  + p \u23a0            \u23a0"
      }
     ],
     "prompt_number": 249
    },
    {
     "cell_type": "markdown",
     "source": "61"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a/(2*sqrt(pi))*exp(-a**2/(4*t))/(t*sqrt(t))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAA7BAMAAAApwt8zAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ne2XsRtsxdXeIARSfiLZjas1lAMVSg/4CGuGTg48quHUAAAAASUVORK5CYII=\n",
       "prompt_number": 250,
       "text": "\n        2   \n      -a    \n      \u2500\u2500\u2500   \n      4\u22c5t   \n   a\u22c5\u212f      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    ___  3/2\n2\u22c5\u2572\u2571 \u03c0 \u22c5t   "
      }
     ],
     "prompt_number": 250
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 251,
       "text": "\n\u239b      \u239b    ___\u239e       \u239b    ___\u239e     \u2502                 \u239b          2      \u239e\u2502   \n\u239c- sinh\u239da\u22c5\u2572\u2571 p \u23a0 + cosh\u239da\u22c5\u2572\u2571 p \u23a0, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 < \n\u239d                                                                             \n\n\u03c0\u239e\n\u2500\u239f\n2\u23a0"
      }
     ],
     "prompt_number": 251
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAVQAAAAYBAMAAABXURglAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\npjg4FgNz6f/cMGII2MzO5jCcbhH2X0I+hA+gfSr/AAAAAElFTkSuQmCC\n",
       "prompt_number": 252,
       "text": "\n                         \u239b    \u239b    ___\u239e            \u239e                          \n- InverseLaplaceTransform\u239dsinh\u239da\u22c5\u2572\u2571 p \u23a0, p, t, None\u23a0 + InverseLaplaceTransform\n\n\u239b    \u239b    ___\u239e            \u239e\n\u239dcosh\u239da\u22c5\u2572\u2571 p \u23a0, p, t, None\u23a0"
      }
     ],
     "prompt_number": 252
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-a*sqrt(p))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAARBAMAAAC/eehCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDLN\n3atI+RUPAAAAy0lEQVQYGWNgwAAc/P////+AIczAMBmLGFgoBYcEpwG6BJNSCEiIzYFTdaE1sqQR\nJ9jGyQxcvAv2QCQYjY2NDVl+ch8AcTMZGNgZeiESYJLrAlMCRzUDZwIDQytDMpIExwPeRs48BuYJ\nDAxpnDeBEoyKJhBpQyErhvkObkBOmLADiGRwgkiASOYHlkCyAMRkK2BA8hHnjwQGBogj+AMVHUAK\noMAcaMXUGyCOfwNUCELNgvPmOzBwwjnIDNYFDBLIfDib01DIAc5BYgAAQhUi8PWAnfsAAAAASUVO\nRK5CYII=\n",
       "prompt_number": 253,
       "text": "\n      ___\n -a\u22c5\u2572\u2571 p \n\u212f        "
      }
     ],
     "prompt_number": 253
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAcYAAAA9BAMAAADFbtR0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8tQFKvHfPKJ/TgmUlKMAAAAASUVORK5CYII=\n",
       "prompt_number": 254,
       "text": "\n                                                 2        2                \n                                               re (a)   im (a)             \n                                             - \u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500             \n  \u239b       \u239bre(a)\u22c5im(a)\u239e      \u239bre(a)\u22c5im(a)\u239e\u239e     4\u22c5t      4\u22c5t               \na\u22c5\u239c- \u2148\u22c5sin\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + cos\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f\u22c5\u212f                 \u22c5Heaviside(t)\n  \u239d       \u239d    2\u22c5t    \u23a0      \u239d    2\u22c5t    \u23a0\u23a0                                \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                    ___  3/2                               \n                                2\u22c5\u2572\u2571 \u03c0 \u22c5t                                  "
      }
     ],
     "prompt_number": 254
    },
    {
     "cell_type": "markdown",
     "source": "62"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-a**2/(4*t))/sqrt(pi*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAA4BAMAAABJf2bZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 255,
       "text": "\n       2   \n     -a    \n     \u2500\u2500\u2500   \n     4\u22c5t   \n    \u212f      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  ___   ___\n\u2572\u2571 \u03c0 \u22c5\u2572\u2571 t "
      }
     ],
     "prompt_number": 255
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 256,
       "text": "\n\u239b      \u239b    ___\u239e       \u239b    ___\u239e                                              \n\u239c- sinh\u239da\u22c5\u2572\u2571 p \u23a0 + cosh\u239da\u22c5\u2572\u2571 p \u23a0     \u2502                 \u239b          2      \u239e\u2502   \n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 < \n\u239c               ___                                                           \n\u239d             \u2572\u2571 p                                                            \n\n \u239e\n\u03c0\u239f\n\u2500\u239f\n2\u239f\n \u23a0"
      }
     ],
     "prompt_number": 256
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_[0], p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXgAAAA3BAMAAAD0ykyYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDg+XLTKB4eHhLYFxW7DZfPBN+A+nJbOpJo8d1AAAAABJRU5ErkJggg==\n",
       "prompt_number": 257,
       "text": "\n                         \u239b    \u239b    ___\u239e            \u239e                          \n                         \u239csinh\u239da\u22c5\u2572\u2571 p \u23a0            \u239f                          \n- InverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f + InverseLaplaceTransform\n                         \u239c      ___                \u239f                          \n                         \u239d    \u2572\u2571 p                 \u23a0                          \n\n\u239b    \u239b    ___\u239e            \u239e\n\u239ccosh\u239da\u22c5\u2572\u2571 p \u23a0            \u239f\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n\u239c      ___                \u239f\n\u239d    \u2572\u2571 p                 \u23a0"
      }
     ],
     "prompt_number": 257
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/sqrt(p)*exp(-a*sqrt(p))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAAwBAMAAABziT7RAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 258,
       "text": "\n      ___\n -a\u22c5\u2572\u2571 p \n\u212f        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    ___  \n  \u2572\u2571 p   "
      }
     ],
     "prompt_number": 258
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAbkAAAA6BAMAAADfKTrZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 259,
       "text": "\n                                               2        2                \n                                             re (a)   im (a)             \n                                           - \u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500             \n\u239b       \u239bre(a)\u22c5im(a)\u239e      \u239bre(a)\u22c5im(a)\u239e\u239e     4\u22c5t      4\u22c5t               \n\u239c- \u2148\u22c5sin\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + cos\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f\u22c5\u212f                 \u22c5Heaviside(t)\n\u239d       \u239d    2\u22c5t    \u23a0      \u239d    2\u22c5t    \u23a0\u23a0                                \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                 ___   ___                               \n                               \u2572\u2571 \u03c0 \u22c5\u2572\u2571 t                                "
      }
     ],
     "prompt_number": 259
    },
    {
     "cell_type": "markdown",
     "source": "63"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "t**n*besselj(n, a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAASBAMAAAD/KA5VAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIjK7iUSZzd12ZqtU\nEO//KStoAAABVElEQVQoFW2RP0jDQBTGv9TEpPnTZlMcNOCgTu3aLUMd3Dq5u4hxEl1chKyCwg26\nRIQOLoqFdnQruLiZSRDBKLhb3BwE372cFxHf8N73fb93d4EA/1Zfp0oFZ3MnOmPhhdo/SGVEviOa\nUl28b8hBdV0OGBF8QXoqRoFlDve4U0uVIIYD0s0ebtHl8FMhPis1MWQ0W0CCXXmj9SFzKo8QFzEs\nAev7CUZYk6n5woCO06xnd8zYdBQAnKGSBc0OtmQDGjnw8wBQ9NXOPOCm2CyZE6I2UQC4D5WcpdMx\njkrmRHDbeocuL+sZWOgFXyWjtzzhM3Bf8YbH0WVMjr65lbvpmBkZezzgHW/HPMb4EPJpO0IjnBkK\nZouAn0S8Y2TdHMGklpKrC1jbg5uQ2RXjqvltMyYXpFXE/+KXxbSwV6RfrUIrrjQrLzRPpfByDZ60\n+iMCoYNz4Bu7IkXxFXUuaAAAAABJRU5ErkJggg==\n",
       "prompt_number": 260,
       "text": "\n n                \nt \u22c5besselj(n, a\u22c5t)"
      }
     ],
     "prompt_number": 260
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 261,
       "text": "\n\u239b                   -n - 1/2                                                  \n\u239c           \u239b     2\u239e                                                          \n\u239c n  -n - 1 \u239c    p \u239f                                                          \n\u239c2 \u22c5a      \u22c5\u239c1 + \u2500\u2500\u239f        \u22c5\u0393(n + 1/2)                                       \n\u239c           \u239c     2\u239f                                                          \n\u239c           \u239d    a \u23a0                                    \u2502                 \u239b   \n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, 0 < n + 1 \u2227 \u2502periodic_argument\u239dpol\n\u239c                  ___                                                        \n\u239d                \u2572\u2571 \u03c0                                                         \n\n                                  \u239e\n                                  \u239f\n                                  \u239f\n                                  \u239f\n                                  \u239f\n       2      \u239e\u2502                  \u239f\nar_lift (a), \u221e\u23a0\u2502 = 0 \u2227 0 < n + 1/2\u239f\n                                  \u239f\n                                  \u23a0"
      }
     ],
     "prompt_number": 261
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simplify(_[0])",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAABBBAMAAAAqBqxsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nOrjafuWW7XuD682ulP4baWKfKRcojVsAAAAASUVORK5CYII=\n",
       "prompt_number": 262,
       "text": "\n                   -n - 1/2           \n           \u239b     2\u239e                   \n n  -n - 1 \u239c    p \u239f                   \n2 \u22c5a      \u22c5\u239c1 + \u2500\u2500\u239f        \u22c5\u0393(n + 1/2)\n           \u239c     2\u239f                   \n           \u239d    a \u23a0                   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                  ___                 \n                \u2572\u2571 \u03c0                  "
      }
     ],
     "prompt_number": 262
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAU8AAABHBAMAAAB1x9gYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 263,
       "text": "\n                                  \u239b        -n - 1/2            \u239e           \n                                  \u239c\u239b     2\u239e                    \u239f           \n n  -n - 1                        \u239c\u239c    p \u239f                    \u239f           \n2 \u22c5a      \u22c5InverseLaplaceTransform\u239c\u239c1 + \u2500\u2500\u239f        , p, t, None\u239f\u22c5\u0393(n + 1/2)\n                                  \u239c\u239c     2\u239f                    \u239f           \n                                  \u239d\u239d    a \u23a0                    \u23a0           \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                     ___                                   \n                                   \u2572\u2571 \u03c0                                    "
      }
     ],
     "prompt_number": 263
    },
    {
     "cell_type": "markdown",
     "source": "64"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "a**n*besseli(n, a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAASBAMAAAD/KA5VAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdmZU3buZ74kiRBAy\nzauYRQn6AAABSElEQVQoFW2RMUvDQBTH/01i0/SaNHQSBak6dsmgsxFxlfYjRIp+DOMmTkEEXcRM\nznHRzUTQUbq46WDHblVw0sH3LpeziA+S+/1/7453IcC/1dVW0/VkOdaWwEt0uqko2MQUOD8YK7Gq\nVqeA6SsW01ZIuK0iOHDNxcCLJMDsNyiJ9ypWZ9sp0FGy7rv3tPNbRY9askb0fqKn1lmjOzYWgDrd\nSVZb6avnCOAwwLDswOoryCs9ILAzGCH2iLjyrlyAM8CQmodbCewYW6q1kSg4pdOsWzzcKnCZii/V\n2lcrblFqg4fTrFFmhEHZ3IFYOpknpmtK7fmmDHZy8ebLPc4HTMvnz3ALSO0GPeARaO72xgnvcYaf\nMXI8ENb8UptRAaxwc7bWcURRhDOu+hdaRXjlwcdaoBn/ckkTHKZEXqYbd5r+gCi/gu0i8AMtCUcd\nfpAt8gAAAABJRU5ErkJggg==\n",
       "prompt_number": 264,
       "text": "\n n                \na \u22c5besseli(n, a\u22c5t)"
      }
     ],
     "prompt_number": 264
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 265,
       "text": "\n\u239b\u23a7             \u239b                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c                                                              \n\u239c\u23aa             \u239c      \u239b                            \u239b            \u239bp\u239e\u239e\u239e         \n\u239c\u23aa             \u239c      \u239c                       p\u22c5sin\u239c(n - 1)\u22c5asin\u239c\u2500\u239f\u239f\u239f         \n\u239c\u23aa             \u239c  ___ \u239c   \u239b            \u239bp\u239e\u239e        \u239d            \u239da\u23a0\u23a0\u239f       __\n\u239c\u23aa             \u239c\u2572\u2571 \u03c0 \u22c5\u239ccos\u239c(n - 1)\u22c5asin\u239c\u2500\u239f\u239f - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f   2\u22c5\u2572\u2571 \u03c0\n\u239c\u23aa             \u239c      \u239c   \u239d            \u239da\u23a0\u23a0              ________   \u239f         \n\u239c\u23aa             \u239c      \u239c                                 \u2571      2    \u239f         \n\u239c\u23aa             \u239c      \u239c                                \u2571      p     \u239f         \n\u239c\u23aa             \u239c      \u239c                          a\u22c5   \u2571   1 - \u2500\u2500    \u239f         \n\u239c\u23aa             \u239c      \u239c                              \u2571         2    \u239f         \n\u239c\u23aa  ___  n + 1 \u239c      \u239d                            \u2572\u2571         a     \u23a0         \n\u239c\u23aa\u2572\u2571 \u03c0 \u22c5a     \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23aa             \u239c                     \u239b-n\u239e  \u239bn    \u239e                            \n\u239c\u23aa             \u239c                    \u0393\u239c\u2500\u2500\u239f\u22c5\u0393\u239c\u2500 + 1\u239f                            \n\u239c\u23aa             \u239d                     \u239d2 \u23a0  \u239d2    \u23a0                            \n\u239c\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23a8                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239d\u23a9                                                                            \n\n    \u239b                                                                         \n    \u239c                                                                         \n    \u239c                                                                         \n    \u239c                                                                         \n    \u239c                                                                         \n    \u239c                                                                         \n    \u239c                                                                         \n    \u239c                                                                         \n    \u239c                                                                         \n    \u239c\u239b                                                              \u239b         \n    \u239c\u239c                                                              \u239c      2\u22c5p\n    \u239c\u239c                                                            3\u22c5\u239c1 + \u2500\u2500\u2500\u2500\u2500\n    \u239c\u239c\u239b        2\u239e \u239b 2 \u239b     2          \u239e    4 \u239b   2          \u239e\u239e     \u239c         \n    \u239c\u239c\u239c     2\u22c5p \u239f \u239cp \u22c5\u239d- 2\u22c5n  + 3\u22c5n + 2\u23a0   p \u22c5\u239d2\u22c5n  - 6\u22c5n + 4\u23a0\u239f     \u239c     2   \n    \u239c\u239c\u239c-2 + \u2500\u2500\u2500\u2500\u239f\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f     \u239c    a \u22c5n\u22c5\n    \u239c\u239c\u239c       2 \u239f \u239c           2                      4        \u239f     \u239c         \n_   \u239c\u239c\u239d      a  \u23a0 \u239d          a                      a         \u23a0     \u239d         \n \u22c5p\u22c5\u239c\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    \u239c\u239c           \u239b 2  2    2\u239e \u239b           2        4\u239e                       2 \n    \u239c\u239c           \u239cn \u22c5p    p \u239f \u239c      4\u22c5n\u22c5p    2\u22c5n\u22c5p \u239f                      n  \n    \u239c\u239c           \u239c\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u239f\u22c5\u239c2\u22c5n - \u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u239f                         \n    \u239c\u239c           \u239c   2     2\u239f \u239c         2        4  \u239f                         \n    \u239c\u239d           \u239d  a     a \u23a0 \u239d        a        a   \u23a0                         \n    \u239d                                                                         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n                                            \u239b                                 \n                                            \u239c                                 \n                                            \u239c                                 \n                                            \u239c                                 \n                                            \u239c                         \u239b      _\n                                            \u239c   \u239b          2        \u239e \u239c     \u2571 \n                            2\u22c5n + 2  -n - 1 \u239c n \u239c       2\u22c5a         \u239f \u239c    \u2571  \n                           a       \u22c5p      \u22c5\u239c2 \u22c5\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + 1\u239f\u22c5\u239c   \u2571   \n                                            \u239c   \u239c     \u239b   2    \u239e    \u239f \u239c  \u2571    \n                                            \u239c   \u239c   2 \u239c2\u22c5a     \u239f    \u239f \u239d\u2572\u2571     \n                                            \u239c   \u239c  p \u22c5\u239c\u2500\u2500\u2500\u2500 - 2\u239f    \u239f         \n                                            \u239c   \u239c     \u239c  2     \u239f    \u239f         \n                                            \u239d   \u239d     \u239d p      \u23a0    \u23a0         \n                           \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                      ___  n +\n                                                                    \u2572\u2571 \u03c0 \u22c5a   \n                                                                              \n                                                                              \n                                                                              \n                                                                    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                                       \u239b                                      \n                                       \u239c                                      \n                                       \u239c                                      \n                                       \u239c                                      \n                                       \u239c                                      \n                                       \u239c                                      \n                                       \u239c\u239b        2\u239e \u239b            2 \u239b     2    \n                                       \u239c\u239c     2\u22c5p \u239f \u239c   2       p \u22c5\u239d- 3\u22c5n  + 6\n                                       \u239c\u239c-2 + \u2500\u2500\u2500\u2500\u239f\u22c5\u239c- n  + 1 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n2          \u239e\u239e                          \u239c\u239c       2 \u239f \u239c                    2    \n \u22c5(n - 2)  \u239f\u239f                          \u239c\u239d      a  \u23a0 \u239d                   a     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f                        p\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239b        2\u239e\u239f\u239f                          \u239c              \u239b 2  2    2\u239e \u239b          \n\u239c     2\u22c5p \u239f\u239f\u239f                          \u239c              \u239cn \u22c5p    p \u239f \u239c      4\u22c5n\u22c5\n\u239c-2 + \u2500\u2500\u2500\u2500\u239f\u239f\u239f                          \u239c              \u239c\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u239f\u22c5\u239c2\u22c5n - \u2500\u2500\u2500\u2500\n\u239c       2 \u239f\u239f\u239f                          \u239c              \u239c   2     2\u239f \u239c         2\n\u239d      a  \u23a0\u23a0\u239f    \u239b            \u239bp\u239e\u239e     \u239d              \u239d  a     a \u23a0 \u239d        a \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5cos\u239c(n - 2)\u22c5asin\u239c\u2500\u239f\u239f + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n            \u239f    \u239d            \u239da\u23a0\u23a0                                            \n- 1         \u239f                                                                 \n            \u239f                                                                 \n            \u239f                                                                 \n            \u23a0                                                                 \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                      \u239bn\u239e  \u239b  n   1\u239e  \u239bn   3\u239e                 \n                                   a\u22c5\u0393\u239c\u2500\u239f\u22c5\u0393\u239c- \u2500 - \u2500\u239f\u22c5\u0393\u239c\u2500 + \u2500\u239f                 \n                                      \u239d2\u23a0  \u239d  2   2\u23a0  \u239d2   2\u23a0                 \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n             2                                                                \n            a                                                                 \n                                                                              \n                                                   -n - 1\u239e                    \n                             \u239b      __________    \u239e      \u239f                    \n                             \u239c     \u2571    2         \u239f      \u239f                    \n              -n    n + 1  2 \u239c    \u2571    a          \u239f      \u239f                    \n_________    \u239e     2     \u22c5a \u22c5\u239c   \u2571   - \u2500\u2500 + 1  + 1\u239f      \u239f                    \n   2         \u239f               \u239c  \u2571       2         \u239f      \u239f                    \n  a          \u239f               \u239d\u2572\u2571       p          \u23a0      \u239f  \u239bn   1\u239e  \u239bn    \u239e  \n- \u2500\u2500 + 1  + 1\u239f   + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u0393\u239c\u2500 + \u2500\u239f\u22c5\u0393\u239c\u2500 + 1\u239f  \n   2         \u239f                    \u239b   2    \u239e             \u239f  \u239d2   2\u23a0  \u239d2    \u23a0  \n  p          \u23a0                  2 \u239c2\u22c5a     \u239f             \u239f                    \n                               p \u22c5\u239c\u2500\u2500\u2500\u2500 - 2\u239f             \u239f                    \n                                  \u239c  2     \u239f             \u239f                    \n                                  \u239d p      \u23a0             \u23a0                    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \n      ___  2                                                                  \n    \u2572\u2571 \u03c0 \u22c5a \u22c5\u0393(n + 1)                                                         \n                                                                              \n           \u239b  n   1  n   1  -n \u2502   \u239e                                          \n           \u239c- \u2500 + \u2500  \u2500 + \u2500, \u2500\u2500 \u2502  2\u239f                                          \n 1 \u256d\u2500\u256e2, 1 \u239c  2   2  2   2  2  \u2502 p \u239f                                          \n  \u22c5\u2502\u2576\u2510     \u239c                   \u2502 \u2500\u2500\u239f                                          \n   \u2570\u2500\u256f3, 3 \u239c            -n     \u2502  2\u239f                                          \n           \u239c0, 1/2      \u2500\u2500     \u2502 a \u239f                                          \n           \u239d            2      \u2502   \u23a0                                          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                          \n             2                                                                \n            a                                                                 \n\n                               \u239b                \u239b         2        \u239e\u239e\u239e        \n                               \u239c                \u239c        p \u22c5(n - 2)\u239f\u239f\u239f        \n                               \u239c  \u239b  n    \u239e   2\u22c5\u239cn - 1 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f\u239f        \n                               \u239c2\u22c5\u239c- \u2500 + 1\u239f     \u239c             2    \u239f\u239f\u239f        \n                               \u239c  \u239d  2    \u23a0     \u239d            a     \u23a0\u239f\u239f        \n                             3\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f        \n  \u239e    4 \u239b   2          \u239e\u239e     \u239c   n - 2            \u239b        2\u239e     \u239f\u239f        \n\u22c5n\u23a0   p \u22c5\u239d2\u22c5n  - 6\u22c5n + 4\u23a0\u239f     \u239c                    \u239c     2\u22c5p \u239f     \u239f\u239f        \n\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f     \u239c                  n\u22c5\u239c-2 + \u2500\u2500\u2500\u2500\u239f     \u239f\u239f        \n                4        \u239f     \u239c                    \u239c       2 \u239f     \u239f\u239f        \n               a         \u23a0     \u239d                    \u239d      a  \u23a0     \u23a0\u239f    \u239b   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5sin\u239c(n \n 2        4\u239e                                   2                     \u239f    \u239d   \np    2\u22c5n\u22c5p \u239f                                  n  - 1                 \u239f        \n\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u239f                                                         \u239f        \n        4  \u239f                                                         \u239f        \n       a   \u23a0                                                         \u23a0        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                         ________                                             \n                        \u2571      2                                              \n                       \u2571      p                                               \n                 a\u22c5   \u2571   1 - \u2500\u2500                                              \n                     \u2571         2                                              \n                   \u2572\u2571         a                                               \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n          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                                                 \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n         \u239bp\u239e\u239e\u239f         \u239f                                                      \n- 2)\u22c5asin\u239c\u2500\u239f\u239f\u239f         \u239f                                                      \n         \u239da\u23a0\u23a0\u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f  \u239bn    \u239e\u239f                                                      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u0393\u239c\u2500 + 1\u239f\u239f                                                      \n             \u239f  \u239d2    \u23a0\u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u239f         \u239f                                                      \n             \u23a0         \u239f                                                      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                                                      \n                       \u239f                                                      \n                       \u239f      \u2502 2\u2502                                            \n                       \u23a0      \u2502p \u2502                                            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  for \u2502\u2500\u2500\u2502 < 1, Max(0, cos(Abs(periodic_argument(polar\n                              \u2502 2\u2502                                            \n                              \u2502a \u2502                                            \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                              \u2502 2\u2502                                            \n                              \u2502a \u2502                                            \n                          for \u2502\u2500\u2500\u2502 < 1                                        \n                              \u2502 2\u2502                                            \n                              \u2502p \u2502                                            \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                           otherwise                                          \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                           n   1   \u2502                 \u239b        \n_lift(a)**2, oo))/2)*sqrt(Abs(a**2))), 0 < \u2500 + \u2500 \u2227 \u2502periodic_argument\u239dpolar_li\n                                           2   2                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                    \u239e\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                   2                \u239f\n  2      \u239e\u2502       p            n    \u239f\nft (a), \u221e\u23a0\u2502 < \u03c0 \u2227 \u2500\u2500 \u2260 1 \u2227 0 < \u2500 + 1\u239f\n                   2           2    \u239f\n                  a                 \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u239f\n                                    \u23a0"
      }
     ],
     "prompt_number": 265
    },
    {
     "cell_type": "markdown",
     "source": "what a mess ...!"
    },
    {
     "cell_type": "markdown",
     "source": "65"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/p*exp(-b*p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACIAAAAtBAMAAAA5P6QkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVKsy\n3c2S+EuhAAAA4UlEQVQoFWNggIPlDnAmlDEdXYBhNYaIqfBJ1YU2yMI3OKJ4F5yDiDAaGxsbchZw\nPWBn6EVSw7GBOaGVIRlJhGfD3AlpnDeBIoyKJhBxIxOGMGEHIDuMwQkiAiQLQCy2AoYUmAjPAxCL\nP1DRAUSDwNQbINK/AUQig/kODJzIfAYG1gUMEqginIZCDqgiA8v7jw4G1jkk2M6lPFF4Aor6ifMV\nmA1QRDrPM7BdQBFhUGVgDUAVCWRgT0AV+cLQjmoyzx+GW6hKGAPFDqCKcCug8hkY2BvQRfxRjWVg\n4HhvCVcDADVCN1LBoFuAAAAAAElFTkSuQmCC\n",
       "prompt_number": 266,
       "text": "\n -b\u22c5p\n\u212f    \n\u2500\u2500\u2500\u2500\u2500\n  p  "
      }
     ],
     "prompt_number": 266
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAI8AAAASBAMAAACQtZA9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nGNjfHtD/lTcyKi4AfwDR06m5cRB5qQAAAABJRU5ErkJggg==\n",
       "prompt_number": 267,
       "text": "Heaviside(-b + t)"
      }
     ],
     "prompt_number": 267
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 268,
       "text": "\n\u239b -b\u22c5p                                              \u239e\n\u239c\u212f         \u239b    1   1    \u239e           \u239b    1        \u239e\u239f\n\u239c\u2500\u2500\u2500\u2500\u2500, 0, \u239c0 < \u2500 \u2228 \u2500 \u2260 0\u239f \u2227 0 < b \u2227 \u239c0 < \u2500 \u2228 0 < b\u239f\u239f\n\u239d  p       \u239d    b   b    \u23a0           \u239d    b        \u23a0\u23a0"
      }
     ],
     "prompt_number": 268
    },
    {
     "cell_type": "markdown",
     "source": "66"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-b*sqrt(p**2+a**2))/sqrt(p**2+a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAA1BAMAAADLxBpJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 269,
       "text": "\n       _________\n      \u2571  2    2 \n -b\u22c5\u2572\u2571  a  + p  \n\u212f               \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     _________  \n    \u2571  2    2   \n  \u2572\u2571  a  + p    "
      }
     ],
     "prompt_number": 269
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKQAAAA3BAMAAABwafyfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\njKz0GDVD+/YdMOH2flPx/3ZLfMB0nOJGAAAAAElFTkSuQmCC\n",
       "prompt_number": 270,
       "text": "\n                       \u239b       _________            \u239e\n                       \u239c      \u2571  2    2             \u239f\n                       \u239c -b\u22c5\u2572\u2571  a  + p              \u239f\n                       \u239c\u212f                           \u239f\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239c     _________              \u239f\n                       \u239c    \u2571  2    2               \u239f\n                       \u239d  \u2572\u2571  a  + p                \u23a0"
      }
     ],
     "prompt_number": 270
    },
    {
     "cell_type": "markdown",
     "source": "67"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-b*sqrt(p**2-a**2))/sqrt(p**2-a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFwAAAA1BAMAAAA+OpH5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
       "prompt_number": 271,
       "text": "\n       ___________\n      \u2571    2    2 \n -b\u22c5\u2572\u2571  - a  + p  \n\u212f                 \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     ___________  \n    \u2571    2    2   \n  \u2572\u2571  - a  + p    "
      }
     ],
     "prompt_number": 271
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAK8AAAA3BAMAAACIiQdoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzHNvEHI9hgAAAABJRU5ErkJggg==\n",
       "prompt_number": 272,
       "text": "\n                       \u239b       ___________            \u239e\n                       \u239c      \u2571    2    2             \u239f\n                       \u239c -b\u22c5\u2572\u2571  - a  + p              \u239f\n                       \u239c\u212f                             \u239f\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239c     ___________              \u239f\n                       \u239c    \u2571    2    2               \u239f\n                       \u239d  \u2572\u2571  - a  + p                \u23a0"
      }
     ],
     "prompt_number": 272
    },
    {
     "cell_type": "markdown",
     "source": "68"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-b*sqrt((p+a)*(p+b)))/sqrt((p+a)*(p+b))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAA1BAMAAAB/4i4DAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 273,
       "text": "\n      _________________\n -b\u22c5\u2572\u2571 (a + p)\u22c5(b + p) \n\u212f                      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    _________________  \n  \u2572\u2571 (a + p)\u22c5(b + p)   "
      }
     ],
     "prompt_number": 273
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAO4AAAA3BAMAAAD58Wv2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 274,
       "text": "\n                       \u239b       ______________________            \u239e\n                       \u239c      \u2571                    2             \u239f\n                       \u239c -b\u22c5\u2572\u2571  a\u22c5b + a\u22c5p + b\u22c5p + p              \u239f\n                       \u239c\u212f                                        \u239f\nInverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                       \u239c     ______________________              \u239f\n                       \u239c    \u2571                    2               \u239f\n                       \u239d  \u2572\u2571  a\u22c5b + a\u22c5p + b\u22c5p + p                \u23a0"
      }
     ],
     "prompt_number": 274
    },
    {
     "cell_type": "markdown",
     "source": "69"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-b*sqrt(p**2+a**2))/(p**2+a**2)*(b+1/sqrt(p**2+a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMgAAABDBAMAAAArAVe2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n15KvHK74ZtiVuCc1/bl++jXc+48XP+lfyV/LCWv5GyoAAAAASUVORK5CYII=\n",
       "prompt_number": 275,
       "text": "\n                          _________\n                         \u2571  2    2 \n\u239b         1      \u239e  -b\u22c5\u2572\u2571  a  + p  \n\u239cb + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u212f               \n\u239c       _________\u239f                 \n\u239c      \u2571  2    2 \u239f                 \n\u239d    \u2572\u2571  a  + p  \u23a0                 \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n               2    2              \n              a  + p               "
      }
     ],
     "prompt_number": 275
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAtsAAAA4BAMAAADeESmeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
       "prompt_number": 276,
       "text": "\n                         \u239b                   1                               \u239e\nb\u22c5InverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                         \u239c         _________            _________            \u239f\n                         \u239c        \u2571  2    2            \u2571  2    2             \u239f\n                         \u239c 2  b\u22c5\u2572\u2571  a  + p      2  b\u22c5\u2572\u2571  a  + p              \u239f\n                         \u239da \u22c5\u212f               + p \u22c5\u212f                          \u23a0\n\n                          \u239b                                1                  \n + InverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                          \u239c                      _________                    \n                          \u239c      _________      \u2571  2    2          _________  \n                          \u239c 2   \u2571  2    2   b\u22c5\u2572\u2571  a  + p      2   \u2571  2    2   \n                          \u239da \u22c5\u2572\u2571  a  + p  \u22c5\u212f               + p \u22c5\u2572\u2571  a  + p  \u22c5\u212f\n\n                          \u239e\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n     _________            \u239f\n    \u2571  2    2             \u239f\nb\u22c5\u2572\u2571  a  + p              \u239f\n                          \u23a0"
      }
     ],
     "prompt_number": 276
    },
    {
     "cell_type": "markdown",
     "source": "70"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-b*sqrt(p**2-a**2))/(p**2-a**2)*(b+1/sqrt(p**2-a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAN4AAABDBAMAAAABsaYZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 277,
       "text": "\n                            ___________\n                           \u2571    2    2 \n\u239b          1       \u239e  -b\u22c5\u2572\u2571  - a  + p  \n\u239cb + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u212f                 \n\u239c       ___________\u239f                   \n\u239c      \u2571    2    2 \u239f                   \n\u239d    \u2572\u2571  - a  + p  \u23a0                   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                  2    2               \n               - a  + p                "
      }
     ],
     "prompt_number": 277
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAz8AAAA4BAMAAAAhlltVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 278,
       "text": "\n                         \u239b                      1                             \nb\u22c5InverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t,\n                         \u239c           ___________            ___________       \n                         \u239c          \u2571    2    2            \u2571    2    2        \n                         \u239c   2  b\u22c5\u2572\u2571  - a  + p      2  b\u22c5\u2572\u2571  - a  + p         \n                         \u239d- a \u22c5\u212f                 + p \u22c5\u212f                       \n\n     \u239e                          \u239b                                     1       \n None\u239f + InverseLaplaceTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     \u239f                          \u239c                          ___________        \n     \u239f                          \u239c        ___________      \u2571    2    2         \n     \u239f                          \u239c   2   \u2571    2    2   b\u22c5\u2572\u2571  - a  + p      2   \n     \u23a0                          \u239d- a \u22c5\u2572\u2571  - a  + p  \u22c5\u212f                 + p \u22c5\u2572\u2571\n\n                                          \u239e\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, p, t, None\u239f\n                   ___________            \u239f\n ___________      \u2571    2    2             \u239f\n\u2571    2    2   b\u22c5\u2572\u2571  - a  + p              \u239f\n  - a  + p  \u22c5\u212f                            \u23a0"
      }
     ],
     "prompt_number": 278
    },
    {
     "cell_type": "markdown",
     "source": "71"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-b*p)-exp(-b*sqrt(p**2+a**2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJQAAAAWBAMAAADUaqg0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nLdzSYT0TUaTdTP+Vfu8cdE278w+MQHg6AH74CgAAAABJRU5ErkJggg==\n",
       "prompt_number": 279,
       "text": "\n         _________        \n        \u2571  2    2         \n   -b\u22c5\u2572\u2571  a  + p      -b\u22c5p\n- \u212f                + \u212f    "
      }
     ],
     "prompt_number": 279
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKkAAAAhBAMAAABQGJapAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEgAAAABJRU5ErkJggg==\n",
       "prompt_number": 280,
       "text": "\n                        \u239b       _________            \u239e\n                        \u239c      \u2571  2    2             \u239f\n                        \u239c -b\u22c5\u2572\u2571  a  + p              \u239f\n-InverseLaplaceTransform\u239d\u212f               , p, t, None\u23a0"
      }
     ],
     "prompt_number": 280
    },
    {
     "cell_type": "markdown",
     "source": "72"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(-b*sqrt(p**2-a**2))-exp(-b*p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJIAAAAVBAMAAABf4KrdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n1GDxF575dTQAyGzcAAAAAElFTkSuQmCC\n",
       "prompt_number": 281,
       "text": "\n       ___________        \n      \u2571    2    2         \n -b\u22c5\u2572\u2571  - a  + p      -b\u22c5p\n\u212f                  - \u212f    "
      }
     ],
     "prompt_number": 281
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATcAAAAhBAMAAABO9PEzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDDb+A8DNTH4MOPi0AAAAAElFTkSuQmCC\n",
       "prompt_number": 282,
       "text": "\n                                                                      \u239b       \n                                                                      \u239c      \u2571\n                         \u239b -b\u22c5p            \u239e                          \u239c -b\u22c5\u2572\u2571 \n- InverseLaplaceTransform\u239d\u212f    , p, t, None\u23a0 + InverseLaplaceTransform\u239d\u212f      \n\n___________            \u239e\n    2    2             \u239f\n - a  + p              \u239f\n           , p, t, None\u23a0"
      }
     ],
     "prompt_number": 282
    },
    {
     "cell_type": "markdown",
     "source": "73"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(1-exp(-a*p))/p",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAqBAMAAAAaH/vSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAABAklEQVQ4EWNgwAnY0maITT/tAJRnVMahaBvjh4LlLA8YGFjCvmMo4dm9e/cejo9A\n6QvMC0CymErAetgXsAUwT2DdgEcJswKXDWuAlAAWJTw5O8CmMOw5tp0zBsJGs2gqQxtECZDkhbJQ\nlbBeYIiAK/FDVsIxEwQSGBj45+Q0wJQwzjKAMFFN6YeKwpSBaVQl8g0MjCjSIA6qEs4ChsNoSjgi\nf21GFmLcc6wBmT9S2f8JgQ8jNWRo4O+yjWc24jeW8eA3ht34lTCLfGCIwq+EgW0Cw0wCSjgPMFwl\noIS/gfkiASX+DlwFBJScPBNCQAWDGiEFDAyXCSrh+EhQid1vAQw1ANPSYDkzN+bAAAAAAElFTkSu\nQmCC\n",
       "prompt_number": 283,
       "text": "\n     -a\u22c5p\n1 - \u212f    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    p    "
      }
     ],
     "prompt_number": 283
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQoAAAASBAMAAABRF424AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nMgPJ+jwsAAAAAElFTkSuQmCC\n",
       "prompt_number": 284,
       "text": "Heaviside(t) - Heaviside(-a + t)"
      }
     ],
     "prompt_number": 284
    },
    {
     "cell_type": "markdown",
     "source": "74"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "(exp(-a*p)-exp(-b*p))/p",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAAtBAMAAABfZVhzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ntIyApyAAAAAASUVORK5CYII=\n",
       "prompt_number": 285,
       "text": "\n   -b\u22c5p    -a\u22c5p\n- \u212f     + \u212f    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n       p       "
      }
     ],
     "prompt_number": 285
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_laplace_transform(_, p, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATYAAAASBAMAAAAj2O8OAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7/b/seOjh9Iu/gMapk+ArEUheQAAAABJRU5ErkJggg==\n",
       "prompt_number": 286,
       "text": "Heaviside(-a + t) - Heaviside(-b + t)"
      }
     ],
     "prompt_number": 286
    },
    {
     "cell_type": "markdown",
     "source": "Other"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "besselj(n, a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADQAAAASBAMAAADmo+DzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZonvmSLNu3ZEMt2r\nEFSC55RrAAABHklEQVQYGU2QMUsDQRCFv9NcjBc2iVpYBERMI4gQBMtAKrGMnVhZB5HgLzgEIYiF\niL3pLbxCG7FIE9FCuFJQyBWClhG0SCE4k92cDszMe99bthj4X15q/pRDvTQydZVe5athkZfYzQ58\njGS25UjebQ4gG6sphg7Zl2KGkCspvOjolHq0i2AAwa6aiiMIob0QTW9ch4wMdy4yEk2FmUiGkJq0\n+XSRX4UymTrFjpAb6cL3OIpgjXzClYJbafnBlqniD+nHzKuvMdeYXDQnW4fqBgQ/lAs8kKi5ZyUu\n5MOiRk1Y9/dy7HsxQYvt2VXoc6lRG85nXo94O5bzlZTAO0u6XnS4soeCTZqhoInTcQBnTvZYHtGn\nNDLdVFrxnPpczC/A9jwA3RDWgQAAAABJRU5ErkJggg==\n",
       "prompt_number": 287,
       "text": "besselj(n, a\u22c5t)"
      }
     ],
     "prompt_number": 287
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 288,
       "text": "\n\u239b                               -n - 1                                        \n\u239c           \u239b      ________    \u239e       \u239b              ________     \u239e          \n\u239c           \u239c     \u2571  2         \u239f       \u239c             \u2571  2          \u239f          \n\u239c n  -n - 1 \u239c    \u2571  a          \u239f       \u239c 2    2     \u2571  a          2\u239f          \n\u239ca \u22c5p      \u22c5\u239c   \u2571   \u2500\u2500 + 1  + 1\u239f      \u22c5\u239ca  + p \u22c5   \u2571   \u2500\u2500 + 1  + p \u239f          \n\u239c           \u239c  \u2571     2         \u239f       \u239c          \u2571     2          \u239f          \n\u239c           \u239d\u2572\u2571     p          \u23a0       \u239d        \u2572\u2571     p           \u23a0         n\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, 0 < \u2500\n\u239c                               2    2                                       2\n\u239d                              a  + p                                         \n\n                                                             \u239e\n                                                             \u239f\n                                                             \u239f\n                                                             \u239f\n                                                             \u239f\n                                                             \u239f\n   1   \u2502                 \u239b          2      \u239e\u2502           n    \u239f\n + \u2500 \u2227 \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 = 0 \u2227 0 < \u2500 + 1\u239f\n   2                                                    2    \u239f\n                                                             \u23a0"
      }
     ],
     "prompt_number": 288
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "besseli(n, a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADMAAAASBAMAAAAEf/uKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIonvq3bduxBUMplm\nzUT5W2AUAAABHUlEQVQYGU2QsUvDQBjFf23TmtSkDQguIg24i39CHdxvdCiSRcTJILgH1MmhmVzN\nIk7SqS4KdhEqOItjdpcKDiIOfpfcRd/wvXe/d3c5Av81qxd/qUJOWFcXOjU2PpUhz8YbOV7JOokh\nxCYsRfCm89geqnYK6Rcw0tVVrqfIKUqDifitzo8G0JfQHl3y/jqVk5oe2Gos4YQ7PaCVgvtjqxXw\nY+7hW0gnBO/DVquyN+KQ7kJXOSxv2eoUBoX7hZ8I0Rd2hpxvP0Wyki9PUj/OHOVVzxjMyPbRdwQ5\nrXB9qIJsDjewdnSMu+jGUrUVvYf5TuhNc3gRIPKSZiTmxjKsyh8lL1HBtUablkMvqrITNvd0clIZ\nlc5sMO6qGuzCL9kcNdO6LaNxAAAAAElFTkSuQmCC\n",
       "prompt_number": 289,
       "text": "besseli(n, a\u22c5t)"
      }
     ],
     "prompt_number": 289
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 290,
       "text": "\n\u239b\u23a7      \u239b                                                                  \u239b  \n\u239c\u23aa      \u239c                                                                  \u239c  \n\u239c\u23aa      \u239c                                                                  \u239c  \n\u239c\u23aa      \u239c                                                                  \u239c  \n\u239c\u23aa      \u239c                                                                  \u239c  \n\u239c\u23aa      \u239c                                                                  \u239c  \n\u239c\u23aa      \u239c                                                                  \u239c  \n\u239c\u23aa      \u239c                                                                  \u239c  \n\u239c\u23aa      \u239c                                                                  \u239c  \n\u239c\u23aa      \u239c                                                                  \u239c\u239b \n\u239c\u23aa      \u239c                                                                  \u239c\u239c \n\u239c\u23aa      \u239c                                                                  \u239c\u239c \n\u239c\u23aa      \u239c                                                                  \u239c\u239c\u239b\n\u239c\u23aa      \u239c                                                                  \u239c\u239c\u239c\n\u239c\u23aa      \u239c      \u239b                            \u239b            \u239bp\u239e\u239e\u239e             \u239c\u239c\u239c\n\u239c\u23aa      \u239c      \u239c                       p\u22c5sin\u239c(n - 1)\u22c5asin\u239c\u2500\u239f\u239f\u239f             \u239c\u239c\u239c\n\u239c\u23aa      \u239c  ___ \u239c   \u239b            \u239bp\u239e\u239e        \u239d            \u239da\u23a0\u23a0\u239f       ___   \u239c\u239c\u239d\n\u239c\u23aa      \u239c\u2572\u2571 \u03c0 \u22c5\u239ccos\u239c(n - 1)\u22c5asin\u239c\u2500\u239f\u239f - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f   2\u22c5\u2572\u2571 \u03c0 \u22c5p\u22c5\u239c\u239c\u2500\n\u239c\u23aa      \u239c      \u239c   \u239d            \u239da\u23a0\u23a0              ________   \u239f             \u239c\u239c \n\u239c\u23aa      \u239c      \u239c                                 \u2571      2    \u239f             \u239c\u239c \n\u239c\u23aa      \u239c      \u239c                                \u2571      p     \u239f             \u239c\u239c \n\u239c\u23aa      \u239c      \u239c                          a\u22c5   \u2571   1 - \u2500\u2500    \u239f             \u239c\u239c \n\u239c\u23aa      \u239c      \u239c                              \u2571         2    \u239f             \u239c\u239d \n\u239c\u23aa  ___ \u239c      \u239d                            \u2572\u2571         a     \u23a0             \u239d  \n\u239c\u23aa\u2572\u2571 \u03c0 \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23aa      \u239c                     \u239b-n\u239e  \u239bn    \u239e                                   \n\u239c\u23aa      \u239c                    \u0393\u239c\u2500\u2500\u239f\u22c5\u0393\u239c\u2500 + 1\u239f                                   \n\u239c\u23aa      \u239d                     \u239d2 \u23a0  \u239d2    \u23a0                                   \n\u239c\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239c\u23a8                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239c\u23aa                                                                            \n\u239d\u23a9                                                                            \n\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                             \u239b         2      \n                                                             \u239c      2\u22c5p \u22c5(n - \n                                                           3\u22c5\u239c1 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        2\u239e \u239b 2 \u239b     2          \u239e    4 \u239b   2          \u239e\u239e     \u239c         \u239b      \n     2\u22c5p \u239f \u239cp \u22c5\u239d- 2\u22c5n  + 3\u22c5n + 2\u23a0   p \u22c5\u239d2\u22c5n  - 6\u22c5n + 4\u23a0\u239f     \u239c     2   \u239c     2\n-2 + \u2500\u2500\u2500\u2500\u239f\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f     \u239c    a \u22c5n\u22c5\u239c-2 + \u2500\n       2 \u239f \u239c           2                      4        \u239f     \u239c         \u239c      \n      a  \u23a0 \u239d          a                      a         \u23a0     \u239d         \u239d      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          \u239b 2  2    2\u239e \u239b           2        4\u239e                       2        \n          \u239cn \u22c5p    p \u239f \u239c      4\u22c5n\u22c5p    2\u22c5n\u22c5p \u239f                      n  - 1    \n          \u239c\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u239f\u22c5\u239c2\u22c5n - \u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u239f                                \n          \u239c   2     2\u239f \u239c         2        4  \u239f                                \n          \u239d  a     a \u23a0 \u239d        a        a   \u23a0                                \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n                                        \u239b                                     \n                                        \u239c                                     \n                                        \u239c                                     \n                                        \u239c                                     \n                                        \u239c                         \u239b      _____\n                                        \u239c   \u239b          2        \u239e \u239c     \u2571    2\n                          n + 1  -n - 1 \u239c n \u239c       2\u22c5a         \u239f \u239c    \u2571    a \n                         a     \u22c5p      \u22c5\u239c2 \u22c5\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + 1\u239f\u22c5\u239c   \u2571   - \u2500\u2500\n                                        \u239c   \u239c     \u239b   2    \u239e    \u239f \u239c  \u2571       2\n                                        \u239c   \u239c   2 \u239c2\u22c5a     \u239f    \u239f \u239d\u2572\u2571       p \n                                        \u239c   \u239c  p \u22c5\u239c\u2500\u2500\u2500\u2500 - 2\u239f    \u239f             \n                                        \u239c   \u239c     \u239c  2     \u239f    \u239f             \n                                        \u239d   \u239d     \u239d p      \u23a0    \u23a0             \n                         \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                      ___ \u256d\u2500\u256e2\n                                                                    \u2572\u2571 \u03c0 \u22c5\u2502\u2576\u2510 \n                                                                          \u2570\u2500\u256f3\n                                                                              \n                                                                              \n                                                                    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                                \u239b                                             \n                                \u239c                                             \n                                \u239c                                             \n                                \u239c                                             \n                                \u239c                                             \n                                \u239c                                             \n                                \u239c\u239b        2\u239e \u239b            2 \u239b     2      \u239e    \n                                \u239c\u239c     2\u22c5p \u239f \u239c   2       p \u22c5\u239d- 3\u22c5n  + 6\u22c5n\u23a0   p\n                                \u239c\u239c-2 + \u2500\u2500\u2500\u2500\u239f\u22c5\u239c- n  + 1 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\n    \u239e\u239e                          \u239c\u239c       2 \u239f \u239c                    2           \n2)  \u239f\u239f                          \u239c\u239d      a  \u23a0 \u239d                   a            \n\u2500\u2500\u2500\u2500\u239f\u239f                        p\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  2\u239e\u239f\u239f                          \u239c              \u239b 2  2    2\u239e \u239b           2     \n\u22c5p \u239f\u239f\u239f                          \u239c              \u239cn \u22c5p    p \u239f \u239c      4\u22c5n\u22c5p    2\u22c5\n\u2500\u2500\u2500\u239f\u239f\u239f                          \u239c              \u239c\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u239f\u22c5\u239c2\u22c5n - \u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\n 2 \u239f\u239f\u239f                          \u239c              \u239c   2     2\u239f \u239c         2       \na  \u23a0\u23a0\u239f    \u239b            \u239bp\u239e\u239e     \u239d              \u239d  a     a \u23a0 \u239d        a        \n\u2500\u2500\u2500\u2500\u2500\u239f\u22c5cos\u239c(n - 2)\u22c5asin\u239c\u2500\u239f\u239f + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     \u239f    \u239d            \u239da\u23a0\u23a0                                                   \n     \u239f                                                                        \n     \u239f                                                                        \n     \u239f                                                                        \n     \u23a0                                                                        \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                               \u239bn\u239e  \u239b  n   1\u239e  \u239bn   3\u239e                        \n                            a\u22c5\u0393\u239c\u2500\u239f\u22c5\u0393\u239c- \u2500 - \u2500\u239f\u22c5\u0393\u239c\u2500 + \u2500\u239f                        \n                               \u239d2\u23a0  \u239d  2   2\u23a0  \u239d2   2\u23a0                        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         a                                                                    \n                                                                              \n                                                                              \n                                               -n - 1\u239e                        \n                         \u239b      __________    \u239e      \u239f                        \n                         \u239c     \u2571    2         \u239f      \u239f                        \n          -n    n + 1  2 \u239c    \u2571    a          \u239f      \u239f                        \n_____    \u239e     2     \u22c5a \u22c5\u239c   \u2571   - \u2500\u2500 + 1  + 1\u239f      \u239f                        \n         \u239f               \u239c  \u2571       2         \u239f      \u239f                        \n         \u239f               \u239d\u2572\u2571       p          \u23a0      \u239f  \u239bn   1\u239e  \u239bn    \u239e      \n + 1  + 1\u239f   + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u0393\u239c\u2500 + \u2500\u239f\u22c5\u0393\u239c\u2500 + 1\u239f      \n         \u239f                    \u239b   2    \u239e             \u239f  \u239d2   2\u23a0  \u239d2    \u23a0      \n         \u23a0                  2 \u239c2\u22c5a     \u239f             \u239f                        \n                           p \u22c5\u239c\u2500\u2500\u2500\u2500 - 2\u239f             \u239f                        \n                              \u239c  2     \u239f             \u239f                        \n                              \u239d p      \u23a0             \u23a0                        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500      \n    ___                                                                       \n  \u2572\u2571 \u03c0 \u22c5a\u22c5\u0393(n + 1)                                                            \n                                                                              \n    \u239b  n   1  n   1  -n \u2502   \u239e                                                 \n    \u239c- \u2500 + \u2500  \u2500 + \u2500, \u2500\u2500 \u2502  2\u239f                                                 \n, 1 \u239c  2   2  2   2  2  \u2502 p \u239f                                                 \n    \u239c                   \u2502 \u2500\u2500\u239f                                                 \n, 3 \u239c            -n     \u2502  2\u239f                                                 \n    \u239c0, 1/2      \u2500\u2500     \u2502 a \u239f                                                 \n    \u239d            2      \u2502   \u23a0                                                 \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                                 \n         a                                                                    \n                                                                              \n\n                        \u239b                \u239b         2        \u239e\u239e\u239e               \n                        \u239c                \u239c        p \u22c5(n - 2)\u239f\u239f\u239f               \n                        \u239c  \u239b  n    \u239e   2\u22c5\u239cn - 1 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f\u239f               \n                        \u239c2\u22c5\u239c- \u2500 + 1\u239f     \u239c             2    \u239f\u239f\u239f               \n                        \u239c  \u239d  2    \u23a0     \u239d            a     \u23a0\u239f\u239f               \n                      3\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f               \n4 \u239b   2          \u239e\u239e     \u239c   n - 2            \u239b        2\u239e     \u239f\u239f               \n \u22c5\u239d2\u22c5n  - 6\u22c5n + 4\u23a0\u239f     \u239c                    \u239c     2\u22c5p \u239f     \u239f\u239f               \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f     \u239c                  n\u22c5\u239c-2 + \u2500\u2500\u2500\u2500\u239f     \u239f\u239f               \n         4        \u239f     \u239c                    \u239c       2 \u239f     \u239f\u239f               \n        a         \u23a0     \u239d                    \u239d      a  \u23a0     \u23a0\u239f    \u239b          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5sin\u239c(n - 2)\u22c5as\n   4\u239e                                   2                     \u239f    \u239d          \nn\u22c5p \u239f                                  n  - 1                 \u239f               \n\u2500\u2500\u2500\u2500\u239f                                                         \u239f               \n 4  \u239f                                                         \u239f               \na   \u23a0                                                         \u23a0               \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                  ________                                                    \n                 \u2571      2                                                     \n                \u2571      p                                                      \n          a\u22c5   \u2571   1 - \u2500\u2500                                                     \n              \u2571         2                                                     \n            \u2572\u2571         a                                                      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n          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                                                 \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n  \u239bp\u239e\u239e\u239f         \u239f                                                             \nin\u239c\u2500\u239f\u239f\u239f         \u239f                                                             \n  \u239da\u23a0\u23a0\u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f  \u239bn    \u239e\u239f                                                             \n\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u0393\u239c\u2500 + 1\u239f\u239f                                                             \n      \u239f  \u239d2    \u23a0\u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u239f         \u239f                                                             \n      \u23a0         \u239f                                                             \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                                                             \n                \u239f                                                             \n                \u239f      \u2502 2\u2502                                                   \n                \u23a0      \u2502p \u2502                                                   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  for \u2502\u2500\u2500\u2502 < 1, Max(0, cos(Abs(periodic_argument(polar_lift(a\n                       \u2502 2\u2502                                                   \n                       \u2502a \u2502                                                   \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                       \u2502 2\u2502                                                   \n                       \u2502a \u2502                                                   \n                   for \u2502\u2500\u2500\u2502 < 1                                               \n                       \u2502 2\u2502                                                   \n                       \u2502p \u2502                                                   \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                    otherwise                                                 \n                                                                              \n                                                                              \n\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                    n   1   \u2502                 \u239b          2    \n)**2, oo))/2)*sqrt(Abs(a**2))), 0 < \u2500 + \u2500 \u2227 \u2502periodic_argument\u239dpolar_lift (a),\n                                    2   2                                     \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                             \u239e\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n            2                \u239f\n  \u239e\u2502       p            n    \u239f\n \u221e\u23a0\u2502 < \u03c0 \u2227 \u2500\u2500 \u2260 1 \u2227 0 < \u2500 + 1\u239f\n            2           2    \u239f\n           a                 \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u239f\n                             \u23a0"
      }
     ],
     "prompt_number": 290
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "erf(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAASBAMAAAANlFvwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdmaZIu8QMt1URIm7\nzauPKR0cAAABJUlEQVQoFV2QMUvDQBiG38b0YmqaXDu4CSWVglNtHdTJWwQX6eYcEFe9H6AkuLll\ndAw4uChkdmrB3yCFLoo/QJGiIAh+SZO76Ad3977v893xccDf6iirVRm5olQ4Vgpgw0dyG0ViSthc\nU6ezTSYqAicFnjT0BGnV7SVA+x90KcsrpP2CVqsr7udrt3uvAvDI19rXuJkOcrO0aR6ZH07sSQJn\ntCbYyTagHiCUuGTf1JfBfcCIsAV8kVkW6Pv+LpsX8ID6U1zBes+gRJ8O9lnAE2CUkDPGlNKzowSW\ngjRQGBhR7HI7H2hljHX1bFOiLg6feTPuAed0u9uSdz/cepuRrnE0hr0XYQ9oPJ+CarGo4irft0gf\nNGykWi+UG6jkVKlSMF4qrAK/awc4LtfHOXIAAAAASUVORK5CYII=\n",
       "prompt_number": 291,
       "text": "erf(a\u22c5t)"
      }
     ],
     "prompt_number": 291
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 292,
       "text": "\n\u239b                    2                                                \u239e\n\u239c                   p                                                 \u239f\n\u239c                  \u2500\u2500\u2500\u2500                                               \u239f\n\u239c                     2                                               \u239f\n\u239c\u239b     \u239b p \u239e    \u239e  4\u22c5a                                                \u239f\n\u239c\u239c- erf\u239c\u2500\u2500\u2500\u239f + 1\u239f\u22c5\u212f                                                   \u239f\n\u239c\u239d     \u239d2\u22c5a\u23a0    \u23a0           \u2502                 \u239b          2      \u239e\u2502   \u03c0\u239f\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, 0, \u2502periodic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 < \u2500\u239f\n\u239d          p                                                         2\u23a0"
      }
     ],
     "prompt_number": 292
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Chi(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADwAAAASBAMAAAD1dKAHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARKuJmVS7EM1m73Yi\nMt1rQmwyAAABOklEQVQoFV2QMUtCYRSGH0u7ifeqVD/ACGooaGlrcQxa7tAWYdAf+KgxKAkiGiKX\nAhcRGoRouLv1EyJxcM2hMUEhyAahcz7z03zhnvOe9+Wc890D00g6Ycz8hhM3HPPyQmMrW2GuNRCW\nrGkIJSheYV/SqmGhi9qJKwlp+SyOYN4wU5SOoW1V7bDoQapEQYdVbbdVl4cmQReCIrda5hmUd2l1\nhIpK/a4d2/mItHhUG754M548wBN7Loq3JYhaG9k/ZBq6wa9ChXheS/gcDjfiZLLWbsMD6ZB3celQ\nyEqKnO1V8XucG+7Vrtkfm7DlNcE3lQTrhPo0T85yAr2/4dzAk3+R4jJpCA71qNsRzf7B5nWzL4vq\nsFQ+fmbvRc5V0g3/cDZRyVGnMSsTRlgckYm85riXc3RMTh1NGX4Bm91TfS5gpmQAAAAASUVORK5C\nYII=\n",
       "prompt_number": 293,
       "text": "Chi(a\u22c5t)"
      }
     ],
     "prompt_number": 293
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "laplace_transform(_, t, p)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 294,
       "text": "\n\u239b\u23a7                     \u239b 2    2\u239e                                              \n\u239c\u23aa                     \u239ca  - p \u239f  \u2148\u22c5\u03c0                                         \n\u239c\u23aa                  log\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u212f                                            \n\u239c\u23aa                     \u239c    2  \u239f                              \u2502 2\u2502            \n\u239c\u23aa                     \u239d   a   \u23a0                              \u2502p \u2502            \n\u239c\u23aa                  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                     for \u2502\u2500\u2500\u2502 < 1, Max(0,\n\u239c\u23aa                         2\u22c5p                                \u2502 2\u2502            \n\u239c\u23aa                                                            \u2502a \u2502            \n\u239c\u23aa                                                                            \n\u239c\u23aa          \u239b     \u239b 2\u239e      \u239b   2    2\u239e\u239e                                      \n\u239c\u23aa          \u239c     \u239ca \u239f      \u239c- a  + p \u239f\u239f  \u2148\u22c5\u03c0                                 \n\u239c\u23aa          \u239c- log\u239c\u2500\u2500\u239f + log\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f\u22c5\u212f                                    \n\u239c\u23aa          \u239c     \u239c 2\u239f      \u239c     2   \u239f\u239f                      \u2502 2\u2502            \n\u239c\u23aa          \u239d     \u239dp \u23a0      \u239d    p    \u23a0\u23a0                      \u2502a \u2502            \n\u239c\u23aa          \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500             for \u2502\u2500\u2500\u2502 < 1        \n\u239c\u23aa                         2\u22c5p                                \u2502 2\u2502            \n\u239c\u23aa                                                            \u2502p \u2502            \n\u239c\u23aa                                                                            \n\u239c\u23aa  \u23a7              \u239b     2\u239e                                                   \n\u239c\u23aa  \u23aa              \u239c    p \u239f                                                   \n\u239c\u23aa  \u23aa         a\u22c5log\u239c1 - \u2500\u2500\u239f                                                   \n\u239c\u23a8  \u23aa              \u239c     2\u239f                \u2502 2\u2502                               \n\u239c\u23aa  \u23aa              \u239d    a \u23a0                \u2502p \u2502      \u2148\u22c5\u03c0                      \n\u239c\u23aa\u03c0\u22c5\u23aa         \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500            for \u2502\u2500\u2500\u2502 < 1\u22c5\u212f                         \n\u239c\u23aa  \u23aa              \u03c0\u22c5p                     \u2502 2\u2502                               \n\u239c\u23aa  \u23aa                                      \u2502a \u2502                               \n\u239c\u23aa  \u23aa                                                                         \n\u239c\u23aa  \u23aa        \u239b 2\u239e        \u239b   2    \u239e                                           \n\u239c\u23aa  \u23aa        \u239ca \u239f        \u239c  a     \u239f                                           \n\u239c\u23aa  \u23a8   a\u22c5log\u239c\u2500\u2500\u239f   a\u22c5log\u239c- \u2500\u2500 + 1\u239f                                           \n\u239c\u23aa  \u23aa        \u239c 2\u239f        \u239c   2    \u239f        \u2502 2\u2502                               \n\u239c\u23aa  \u23aa        \u239dp \u23a0        \u239d  p     \u23a0        \u2502a \u2502                               \n\u239c\u23aa  \u23aa - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500    for \u2502\u2500\u2500\u2502 < 1                           \n\u239c\u23aa  \u23aa      \u03c0\u22c5p            \u03c0\u22c5p              \u2502 2\u2502                               \n\u239c\u23aa  \u23aa                                      \u2502p \u2502                               \n\u239c\u23aa  \u23aa                                                                         \n\u239c\u23aa  \u23aa        \u239b                  \u2502  2\u239e                                         \n\u239c\u23aa  \u23aa\u256d\u2500\u256e2, 2 \u239c1/2, 1/2   0, 0   \u2502 p \u239f                                         \n\u239c\u23aa  \u23aa\u2502\u2576\u2510     \u239c                  \u2502 \u2500\u2500\u239f   otherwise                             \n\u239c\u23aa  \u23aa\u2570\u2500\u256f4, 4 \u239c 0, 1/2   0, -1/2 \u2502  2\u239f                                         \n\u239c\u23aa  \u23a9        \u239d                  \u2502 a \u23a0                                         \n\u239c\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   otherwise          \n\u239d\u23a9                         2\u22c5a                                                \n\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                       \u2502      \n cos(Abs(periodic_argument(polar_lift(a)**2, oo))/2)*sqrt(Abs(a**2))), \u2502period\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\n                                            \u239e\n                                            \u239f\n                                            \u239f\n                                       2    \u239f\n           \u239b          2      \u239e\u2502       p     \u239f\nic_argument\u239dpolar_lift (a), \u221e\u23a0\u2502 < \u03c0 \u2227 \u2500\u2500 \u2260 1\u239f\n                                       2    \u239f\n                                      a     \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u239f\n                                            \u23a0"
      }
     ],
     "prompt_number": 294
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "_[0]",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": 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39SsuDE8WXZSUNzGDLhF7kn3rvm6CbuTeBXSZ92BaGbz+6M8mvM4AlWQne8h/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\nA7Q09SAs8D8ChZATjt+WSQAAAABJRU5ErkJggg==\n",
       "prompt_number": 295,
       "text": "\n\u23a7                     \u239b 2    2\u239e                                      \n\u23aa                     \u239ca  - p \u239f  \u2148\u22c5\u03c0                                 \n\u23aa                  log\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u212f                                    \n\u23aa                     \u239c    2  \u239f                              \u2502 2\u2502    \n\u23aa                     \u239d   a   \u23a0                              \u2502p \u2502    \n\u23aa                  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                     for \u2502\u2500\u2500\u2502 < 1\n\u23aa                         2\u22c5p                                \u2502 2\u2502    \n\u23aa                                                            \u2502a \u2502    \n\u23aa                                                                    \n\u23aa          \u239b     \u239b 2\u239e      \u239b   2    2\u239e\u239e                              \n\u23aa          \u239c     \u239ca \u239f      \u239c- a  + p \u239f\u239f  \u2148\u22c5\u03c0                         \n\u23aa          \u239c- log\u239c\u2500\u2500\u239f + log\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f\u22c5\u212f                            \n\u23aa          \u239c     \u239c 2\u239f      \u239c     2   \u239f\u239f                      \u2502 2\u2502    \n\u23aa          \u239d     \u239dp \u23a0      \u239d    p    \u23a0\u23a0                      \u2502a \u2502    \n\u23aa          \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500             for \u2502\u2500\u2500\u2502 < 1\n\u23aa                         2\u22c5p                                \u2502 2\u2502    \n\u23aa                                                            \u2502p \u2502    \n\u23aa                                                                    \n\u23aa  \u23a7              \u239b     2\u239e                                           \n\u23aa  \u23aa              \u239c    p \u239f                                           \n\u23aa  \u23aa         a\u22c5log\u239c1 - \u2500\u2500\u239f                                           \n\u23a8  \u23aa              \u239c     2\u239f                \u2502 2\u2502                       \n\u23aa  \u23aa              \u239d    a \u23a0                \u2502p \u2502      \u2148\u22c5\u03c0              \n\u23aa\u03c0\u22c5\u23aa         \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500            for \u2502\u2500\u2500\u2502 < 1\u22c5\u212f                 \n\u23aa  \u23aa              \u03c0\u22c5p                     \u2502 2\u2502                       \n\u23aa  \u23aa                                      \u2502a \u2502                       \n\u23aa  \u23aa                                                                 \n\u23aa  \u23aa        \u239b 2\u239e        \u239b   2    \u239e                                   \n\u23aa  \u23aa        \u239ca \u239f        \u239c  a     \u239f                                   \n\u23aa  \u23a8   a\u22c5log\u239c\u2500\u2500\u239f   a\u22c5log\u239c- \u2500\u2500 + 1\u239f                                   \n\u23aa  \u23aa        \u239c 2\u239f        \u239c   2    \u239f        \u2502 2\u2502                       \n\u23aa  \u23aa        \u239dp \u23a0        \u239d  p     \u23a0        \u2502a \u2502                       \n\u23aa  \u23aa - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500    for \u2502\u2500\u2500\u2502 < 1                   \n\u23aa  \u23aa      \u03c0\u22c5p            \u03c0\u22c5p              \u2502 2\u2502                       \n\u23aa  \u23aa                                      \u2502p \u2502                       \n\u23aa  \u23aa                                                                 \n\u23aa  \u23aa        \u239b                  \u2502  2\u239e                                 \n\u23aa  \u23aa\u256d\u2500\u256e2, 2 \u239c1/2, 1/2   0, 0   \u2502 p \u239f                                 \n\u23aa  \u23aa\u2502\u2576\u2510     \u239c                  \u2502 \u2500\u2500\u239f   otherwise                     \n\u23aa  \u23aa\u2570\u2500\u256f4, 4 \u239c 0, 1/2   0, -1/2 \u2502  2\u239f                                 \n\u23aa  \u23a9        \u239d                  \u2502 a \u23a0                                 \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   otherwise  \n\u23a9                         2\u22c5a                                        "
      }
     ],
     "prompt_number": 295
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "",
     "language": "python",
     "outputs": [],
     "prompt_number": 295
    }
   ]
  }
 ]
}