npm package: @stdlib/math-base-special-spence
Description: Spence's function, also known as the dilogarithm.
Author: The Stdlib Authors
Latest version: 0.3.1

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# spence

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

> [Spence's function][spence], also known as the dilogarithm.

<section class="intro">

The dilogarithm is defined as

<!-- <equation class="equation" label="eq:dilogarithm" align="center" raw="\operatorname{Li}_{2}(z) = -\int_{0}^{z}{\ln(1-u) \over u}\,du{\text{, }}z\in \mathbb {C}" alt="Dilogarithm."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Li}_{2}(z) = -\int_{0}^{z}{\ln(1-u) \over u}\,du{\text{, }}z\in \mathbb {C}" data-equation="eq:dilogarithm">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@ea8657e10f42753f63de0b5c7dd8b13b4879409a/lib/node_modules/@stdlib/math/base/special/spence/docs/img/equation_dilogarithm.svg" alt="Dilogarithm.">
    <br>
</div> -->

<!-- </equation> -->

or also alternatively as

<!-- <equation class="equation" label="eq:dilogarithm_alt" align="center" raw="\int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname {Li} _{2}(1-v)." alt="Alternative definition of dilogarithm."> -->

<!-- <div class="equation" align="center" data-raw-text="\int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname {Li} _{2}(1-v)." data-equation="eq:dilogarithm_alt">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@ea8657e10f42753f63de0b5c7dd8b13b4879409a/lib/node_modules/@stdlib/math/base/special/spence/docs/img/equation_dilogarithm_alt.svg" alt="Alternative definition of dilogarithm.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/math-base-special-spence
```

</section>

<section class="usage">

## Usage

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

#### spence( x )

Evaluates [Spence's function][spence], which is alternatively known as the dilogarithm.

```javascript
var v = spence( 3.0 );
// returns ~-1.437

v = spence( 0.0 );
// returns ~1.645

v = spence( NaN );
// returns NaN
```

For negative numbers, the dilogarithm is **not** defined.

```javascript
var v = spence( -4.0 );
// 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 spence = require( '@stdlib/math-base-special-spence' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 100, 0.0, 100.0, opts );

logEachMap( 'spence( %0.4f ) = %0.4f', x, spence );
```

</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/spence.h"
```

#### stdlib_base_spence( x )

Evaluates [Spence's function][spence], which is alternatively known as the dilogarithm.

```c
double out = stdlib_base_spence( 3.0 );
// returns ~-1.437

out = stdlib_base_spence( 0.0 );
// returns ~1.645
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.

```c
double stdlib_base_spence( const double x );
```

</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/spence.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 3.0, 9.0, 0.0, -10.0 };

    double y;
    int i;
    for ( i = 0; i < 4; i++ ) {
        y = stdlib_base_spence( x[ i ] );
        printf( "spence(%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="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

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

---

## Copyright

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

</section>

<!-- /.stdlib -->

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

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

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

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

[spence]: https://en.wikipedia.org/wiki/Spence%27s_function

</section>

<!-- /.links -->