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https://github.com/stdlib-js/math-base-special-spence

Spence’s function, also known as the dilogarithm.
https://github.com/stdlib-js/math-base-special-spence

dilogarithm function integral javascript math mathematics node node-js nodejs number polylogarithm special spence stdlib

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Spence’s function, also known as the dilogarithm.

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README 

        



  

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# spence



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



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



The dilogarithm is defined as



```math

\mathop{\mathrm{Li}}_{2}(z) = -\int_{0}^{z}{\ln(1-u) \over u}\,du{\text{, }}z\in \mathbb {C}

```



or also alternatively as



```math

\int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname {Li} _{2}(1-v).

```



## Installation



```bash

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

```



Alternatively,



-   To load the package in a website via a `script` tag without installation and bundlers, use the [ES Module][es-module] available on the [`esm`][esm-url] branch (see [README][esm-readme]).

-   If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions).

-   For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [`umd`][umd-url] branch (see [README][umd-readme]).



The [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships.



To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.



## 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

```



## Examples



```javascript

var randu = require( '@stdlib/random-base-randu' );

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



var x;

var i;



for ( i = 0; i < 100; i++ ) {

    x = randu() * 100.0;

    console.log( 'spence( %d ) = %d', x, spence( x ) );

}

```



* * *



## C APIs



### 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 );

```



### Examples



```c

#include "stdlib/math/base/special/spence.h"

#include 



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 );

    }

}

```



* * *



## 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 © 2016-2024. The Stdlib [Authors][stdlib-authors].



[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=main

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



[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/math-base-special-spence/main.svg

[coverage-url]: https://codecov.io/github/stdlib-js/math-base-special-spence?branch=main



[chat-image]: https://img.shields.io/gitter/room/stdlib-js/stdlib.svg

[chat-url]: https://app.gitter.im/#/room/#stdlib-js_stdlib:gitter.im



[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-spence/tree/deno

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

[umd-url]: https://github.com/stdlib-js/math-base-special-spence/tree/umd

[umd-readme]: https://github.com/stdlib-js/math-base-special-spence/blob/umd/README.md

[esm-url]: https://github.com/stdlib-js/math-base-special-spence/tree/esm

[esm-readme]: https://github.com/stdlib-js/math-base-special-spence/blob/esm/README.md

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



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