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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:

where ζ(s) is the Riemann zeta function.


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


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


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 ] ) );


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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