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Dirichlet Eta Function

Dirichlet eta function.

The Dirichlet eta function is defined by the Dirichlet series

$$\eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots$$

where s is a complex variable equal to σ + ti. The series is convergent for all complex numbers having a real part greater than 0.

Note that the Dirichlet eta function is also known as the alternating zeta function and denoted ζ*(s). The series is an alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function. Accordingly, the following relation holds:

$$\eta(s) = (1-2^{1-s})\zeta(s)$$

where ζ(s) is the Riemann zeta function.

Installation

npm install @stdlib/math-base-special-dirichlet-eta

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var eta = require( '@stdlib/math-base-special-dirichlet-eta' );

eta( s )

Evaluates the Dirichlet eta function as a function of a real variable s.

var v = eta( 0.0 ); // Abel sum of 1-1+1-1+...
// returns 0.5

v = eta( -1.0 ); // Abel sum of 1-2+3-4+...
// returns 0.25

v = eta( 1.0 ); // alternating harmonic series => ln(2)
// returns 0.6931471805599453

v = eta( 3.14 );
// returns ~0.9096

v = eta( NaN );
// returns NaN

Examples

var linspace = require( '@stdlib/array-base-linspace' );
var eta = require( '@stdlib/math-base-special-dirichlet-eta' );

var s = linspace( -50.0, 50.0, 200 );

var i;
for ( i = 0; i < s.length; i++ ) {
    console.log( 's: %d, η(s): %d', s[ i ], eta( s[ i ] ) );
}

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

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