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# Dirichlet Eta Function

> [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="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 -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

[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://github.com/stdlib-js/math/tree/main/base/special/riemann-zeta

</section>

<!-- /.links -->
