@stdlib/math-base-special-dirichlet-eta
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    0.0.5 • Public • Published

    Dirichlet Eta Function

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    Dirichlet eta function.

    The Dirichlet eta function is defined by the Dirichlet series

    Dirichlet eta function

    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:

    Dirichlet-Riemann zeta relation

    where ζ(s) is the Riemann zeta function.

    Installation

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

    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-linspace' );
    var eta = require( '@stdlib/math-base-special-dirichlet-eta' );
    
    var s = linspace( -50.0, 50.0, 200 );
    var v;
    var i;
    
    for ( i = 0; i < s.length; i++ ) {
        v = eta( s[ i ] );
        console.log( 's: %d, η(s): %d', s[ i ], v );
    }

    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.

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    License

    See LICENSE.

    Copyright

    Copyright © 2016-2021. The Stdlib Authors.

    Install

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

    Homepage

    stdlib.io

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    11

    Version

    0.0.5

    License

    Apache-2.0

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    53.7 kB

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    11

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    Collaborators

    • stdlib-bot
    • kgryte
    • planeshifter
    • rreusser