Dirichlet Eta Function

Dirichlet eta function.

The Dirichlet eta function is defined by the Dirichlet series

eta left-parenthesis s right-parenthesis equals sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts StartFraction left-parenthesis negative 1 right-parenthesis Superscript n minus 1 Baseline Over n Superscript s Baseline EndFraction equals StartFraction 1 Over 1 Superscript s Baseline EndFraction minus StartFraction 1 Over 2 Superscript s Baseline EndFraction plus StartFraction 1 Over 3 Superscript s Baseline EndFraction minus StartFraction 1 Over 4 Superscript s Baseline EndFraction plus midline-horizontal-ellipsis

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 left-parenthesis s right-parenthesis equals left-parenthesis 1 minus 2 Superscript 1 minus s Baseline right-parenthesis zeta left-parenthesis s right-parenthesis

where ζ(s) is the Riemann zeta function.

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/math/utils/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 );
}