math-dirichlet-eta

Dirichlet Eta Function
===
[![NPM version][npm-image]][npm-url] [![Build Status][build-image]][build-url] [![Coverage Status][coverage-image]][coverage-url] [![Dependencies][dependencies-image]][dependencies-url]

> [Dirichlet eta][eta-function] function.

The [Dirichlet eta][eta-function] function is defined by the [Dirichlet series][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][eta-function] 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][zeta-function] function. Accordingly, the following relation holds:

where ζ(s) is the [Riemann zeta][zeta-function] function.

Installation

`` bash
$ npm install math-dirichlet-eta
`

Usage

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

#### eta( s )

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

` javascript
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

` javascript
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,

` bash
$ node ./examples/index.js
`


---


Tests

$3

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

` bash
$ make test
`

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

$3

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

` bash
$ make test-cov
`

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

` bash
$ make view-cov
`

$3

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

` bash
$ make test-browsers
`

To view the tests in a local web browser,

` bash
$ make view-browser-tests
``

---

License

MIT license.

Copyright

Copyright © 2016. The [Compute.io][compute-io] Authors.

[npm-image]: http://img.shields.io/npm/v/math-dirichlet-eta.svg
[npm-url]: https://npmjs.org/package/math-dirichlet-eta

[build-image]: http://img.shields.io/travis/math-io/dirichlet-eta/master.svg
[build-url]: https://travis-ci.org/math-io/dirichlet-eta

[coverage-image]: https://img.shields.io/codecov/c/github/math-io/dirichlet-eta/master.svg
[coverage-url]: https://codecov.io/github/math-io/dirichlet-eta?branch=master

[dependencies-image]: http://img.shields.io/david/math-io/dirichlet-eta.svg
[dependencies-url]: https://david-dm.org/math-io/dirichlet-eta

[dev-dependencies-image]: http://img.shields.io/david/dev/math-io/dirichlet-eta.svg
[dev-dependencies-url]: https://david-dm.org/dev/math-io/dirichlet-eta

[github-issues-image]: http://img.shields.io/github/issues/math-io/dirichlet-eta.svg
[github-issues-url]: https://github.com/math-io/dirichlet-eta/issues

[tape]: https://github.com/substack/tape
[istanbul]: https://github.com/gotwarlost/istanbul
[testling]: https://ci.testling.com

[compute-io]: https://github.com/compute-io/
[eta-function]: https://en.wikipedia.org/wiki/Dirichlet_eta_function
[dirichlet-series]: https://en.wikipedia.org/wiki/Dirichlet_series
[zeta-function]: https://github.com/math-io/riemann-zeta