Dirichlet Eta Function

Dirichlet eta function.

The Dirichlet eta function is defined by the Dirichlet series

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.

Installation

$ npm install math-dirichlet-eta

Usage

var eta = require( 'math-dirichlet-eta' );

eta( s )

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

var v = eta( 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
// returns 0.6931471805599453 => ln(2)

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

v = eta( NaN );
// returns NaN

Examples

var linspace = require( 'compute-linspace' );
var eta = require( 'math-dirichlet-eta' );

var s = linspace( -50, 50, 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 );
}

To run the example code from the top-level application directory,

$ node ./examples/index.js

Tests

Unit

This repository uses tape for unit tests. To run the tests, execute the following command in the top-level application directory:

$ make test

All new feature development should have corresponding unit tests to validate correct functionality.

Test Coverage

This repository uses Istanbul as its code coverage tool. To generate a test coverage report, execute the following command in the top-level application directory:

$ make test-cov

Istanbul creates a ./reports/coverage directory. To access an HTML version of the report,

$ make view-cov

Browser Support

This repository uses Testling for browser testing. To run the tests in a (headless) local web browser, execute the following command in the top-level application directory:

$ make test-browsers

To view the tests in a local web browser,

$ make view-browser-tests

License

MIT license.

Copyright

Copyright © 2016. The Compute.io Authors.