npm package: @stdlib/math-base-special-dirichlet-eta
Description: Dirichlet eta function.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Dirichlet Eta Function

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

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

<section class="intro">

The [Dirichlet eta][eta-function] function is defined by the [Dirichlet series][dirichlet-series]

<!-- <equation class="equation" label="eq:dirichlet_eta_function" align="center" raw="\eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots" alt="Dirichlet eta function"> -->

<div class="equation" align="center" data-raw-text="\eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots" data-equation="eq:dirichlet_eta_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/dirichlet-eta/docs/img/equation_dirichlet_eta_function.svg" alt="Dirichlet eta function">
    <br>
</div> -->

<!-- </equation> -->

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:

<!-- <equation class="equation" label="eq:dirichlet_riemann_relation" align="center" raw="\eta(s) = (1-2^{1-s})\zeta(s)" alt="Dirichlet-Riemann zeta relation"> -->

<!-- <div class="equation" align="center" data-raw-text="\eta(s) = (1-2^{1-s})\zeta(s)" data-equation="eq:dirichlet_riemann_relation">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/dirichlet-eta/docs/img/equation_dirichlet_riemann_relation.svg" alt="Dirichlet-Riemann zeta relation">
    <br>
</div>

<!-- </equation> -->

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

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## 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
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```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 );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```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 );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

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

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 );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## 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

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

</section>

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[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

</section>

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