About stdlib...

We believe in a future in which the web is a preferred environment for numerical computation. To help realize this future, we've built stdlib. stdlib is a standard library, with an emphasis on numerical and scientific computation, written in JavaScript (and C) for execution in browsers and in Node.js.

The library is fully decomposable, being architected in such a way that you can swap out and mix and match APIs and functionality to cater to your exact preferences and use cases.

When you use stdlib, you can be absolutely certain that you are using the most thorough, rigorous, well-written, studied, documented, tested, measured, and high-quality code out there.

To join us in bringing numerical computing to the web, get started by checking us out on GitHub, and please consider financially supporting stdlib. We greatly appreciate your continued support!

Dirichlet Eta Function

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url]

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

The [Dirichlet eta][eta-function] function is defined by the [Dirichlet series][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][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][@stdlib/math/base/special/riemann-zeta] function. Accordingly, the following relation holds:

where ζ(s) is the [Riemann zeta][@stdlib/math/base/special/riemann-zeta] function.

Installation

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

`Usage`

`javascript var eta = require( '@stdlib/math-base-special-dirichlet-eta' );`

#### eta( s )

Evaluates the Dirichlet eta function for a double-precision floating-point number s.

`javascript 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`

`javascript var uniform = require( '@stdlib/random-array-uniform' ); var logEachMap = require( '@stdlib/console-log-each-map' ); var eta = require( '@stdlib/math-base-special-dirichlet-eta' );

var opts = { 'dtype': 'float64' }; var s = uniform( 200, -50.0, 50.0, opts );

logEachMap( 's: %0.4f, η(s): %0.4f', s, eta );`

*

`C APIs`

`$3`

`c #include "stdlib/math/base/special/dirichlet_eta.h"`

#### stdlib_base_eta( s )

Evaluates the Dirichlet eta function for a double-precision floating-point number s.

`c double y = stdlib_base_eta( 0.0 ); // returns 0.5`

The function accepts the following arguments:

- s: [in] double input value.

`c double stdlib_base_eta( const double s );`

`$3`

`c #include "stdlib/math/base/special/dirichlet_eta.h" #include

int main( void ) { const double x[] = { 45.0, 90.0, 0.0, 0.0 / 0.0 };

double y; int i; for ( i = 0; i < 4; i++ ) { y = stdlib_base_eta( x[ i ] ); printf( "η(%lf) = %lf\n", x[ i ], y ); } }`

*

`Notice`

This package is part of [stdlib][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][stdlib], see the main project [repository][stdlib].

#### Community

[![Chat][chat-image]][chat-url]

---

`License`

See [LICENSE][stdlib-license].

`Copyright`

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

[test-image]: https://github.com/stdlib-js/math-base-special-dirichlet-eta/actions/workflows/test.yml/badge.svg?branch=v0.3.1 [test-url]: https://github.com/stdlib-js/math-base-special-dirichlet-eta/actions/workflows/test.yml?query=branch:v0.3.1

[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/math-base-special-dirichlet-eta/main.svg [coverage-url]: https://codecov.io/github/stdlib-js/math-base-special-dirichlet-eta?branch=main

[chat-image]: https://img.shields.io/badge/zulip-join_chat-brightgreen.svg [chat-url]: https://stdlib.zulipchat.com

[stdlib]: https://github.com/stdlib-js/stdlib

[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors

[umd]: https://github.com/umdjs/umd [es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules

[deno-url]: https://github.com/stdlib-js/math-base-special-dirichlet-eta/tree/deno [deno-readme]: https://github.com/stdlib-js/math-base-special-dirichlet-eta/blob/deno/README.md [umd-url]: https://github.com/stdlib-js/math-base-special-dirichlet-eta/tree/umd [umd-readme]: https://github.com/stdlib-js/math-base-special-dirichlet-eta/blob/umd/README.md [esm-url]: https://github.com/stdlib-js/math-base-special-dirichlet-eta/tree/esm [esm-readme]: https://github.com/stdlib-js/math-base-special-dirichlet-eta/blob/esm/README.md [branches-url]: https://github.com/stdlib-js/math-base-special-dirichlet-eta/blob/main/branches.md

[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-special-dirichlet-eta/main/LICENSE

[eta-function]: https://en.wikipedia.org/wiki/Dirichlet_eta_function

[dirichlet-series]: https://en.wikipedia.org/wiki/Dirichlet_series

[@stdlib/math/base/special/riemann-zeta]: https://www.npmjs.com/package/@stdlib/math-base-special-riemann-zeta


  
    About stdlib...
  

  We believe in a future in which the web is a preferred environment for numerical computation. To help realize this future, we've built stdlib. stdlib is a standard library, with an emphasis on numerical and scientific computation, written in JavaScript (and C) for execution in browsers and in Node.js.

  The library is fully decomposable, being architected in such a way that you can swap out and mix and match APIs and functionality to cater to your exact preferences and use cases.

  When you use stdlib, you can be absolutely certain that you are using the most thorough, rigorous, well-written, studied, documented, tested, measured, and high-quality code out there.

  To join us in bringing numerical computing to the web, get started by checking us out on GitHub, and please consider financially supporting stdlib. We greatly appreciate your continued support!

`Dirichlet Eta Function`

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-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][@stdlib/math/base/special/riemann-zeta] function. Accordingly, the following relation holds:
where ζ(s) is the [Riemann zeta][@stdlib/math/base/special/riemann-zeta] function.

`Installation`

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

`Usage`

`javascript var eta = require( '@stdlib/math-base-special-dirichlet-eta' ); `

#### eta( s )

Evaluates the Dirichlet eta function for a double-precision floating-point number s.

`javascript 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`

`javascript var uniform = require( '@stdlib/random-array-uniform' ); var logEachMap = require( '@stdlib/console-log-each-map' ); var eta = require( '@stdlib/math-base-special-dirichlet-eta' );

var opts = { 'dtype': 'float64' }; var s = uniform( 200, -50.0, 50.0, opts );

logEachMap( 's: %0.4f, η(s): %0.4f', s, eta ); `

*

`C APIs`

`$3`

`c #include "stdlib/math/base/special/dirichlet_eta.h" `

#### stdlib_base_eta( s )

Evaluates the Dirichlet eta function for a double-precision floating-point number s.

`c double y = stdlib_base_eta( 0.0 ); // returns 0.5 `

The function accepts the following arguments:

- s: [in] double input value.

`c double stdlib_base_eta( const double s ); `

`$3`

`c #include "stdlib/math/base/special/dirichlet_eta.h" #include

int main( void ) { const double x[] = { 45.0, 90.0, 0.0, 0.0 / 0.0 };

double y; int i; for ( i = 0; i < 4; i++ ) { y = stdlib_base_eta( x[ i ] ); printf( "η(%lf) = %lf\n", x[ i ], y ); } } `

*

`Notice`

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

#### Community

[![Chat][chat-image]][chat-url]

---

`License`

See [LICENSE][stdlib-license].

`Copyright`

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

[chat-image]: https://img.shields.io/badge/zulip-join_chat-brightgreen.svg [chat-url]: https://stdlib.zulipchat.com

[stdlib]: https://github.com/stdlib-js/stdlib

[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors

[umd]: https://github.com/umdjs/umd [es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules

[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-special-dirichlet-eta/main/LICENSE

[eta-function]: https://en.wikipedia.org/wiki/Dirichlet_eta_function

[dirichlet-series]: https://en.wikipedia.org/wiki/Dirichlet_series

[@stdlib/math/base/special/riemann-zeta]: https://www.npmjs.com/package/@stdlib/math-base-special-riemann-zeta