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

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</details>

# Gamma Function

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

> [Gamma][gamma-function] function.

<section class="intro">

The [gamma function][gamma-function] extends the [factorial function][@stdlib/math/base/special/factorial] to [real][real] and [complex][complex] numbers. If `n` is a positive `integer`,

<!-- <equation class="equation" label="eq:gamma_function_positive_integers" align="center" raw="\Gamma ( n ) = (n-1)!" alt="Gamma function for positive integers."> -->

<div class="equation" align="center" data-raw-text="\Gamma ( n ) = (n-1)!" data-equation="eq:gamma_function_positive_integers">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gamma/docs/img/equation_gamma_function_positive_integers.svg" alt="Gamma function for positive integers.">
    <br>
</div> -->

<!-- </equation> -->

Generalized to all complex numbers `z`, except for nonpositive integers, the [gamma function][gamma-function] can be expressed as an infinite product

<!-- <equation class="equation" label="eq:gamma_function_infinite_product" align="center" raw="\Gamma ( z ) = \frac{e^{-\gamma z}}{z} \prod^{\infty}_{n=1} \left ( 1+\frac{z}{n}\right )^{-1} e^{z/n}" alt="Gamma function for all complex numbers."> -->

<!-- <div class="equation" align="center" data-raw-text="\Gamma ( z ) = \frac{e^{-\gamma z}}{z} \prod^{\infty}_{n=1} \left ( 1+\frac{z}{n}\right )^{-1} e^{z/n}" data-equation="eq:gamma_function_infinite_product">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/gamma/docs/img/equation_gamma_function_infinite_product.svg" alt="Gamma function for all complex numbers.">
    <br>
</div>

<!-- </equation> -->

where `γ ≈ 0.577216` is the  [Euler–Mascheroni constant][@stdlib/constants/float64/eulergamma].

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### gamma( x )

Evaluates the [gamma function][gamma-function].

```javascript
var v = gamma( 4.0 );
// returns 6.0

v = gamma( -1.5 );
// returns ~2.363

v = gamma( -0.5 );
// returns ~-3.545

v = gamma( 0.5 );
// returns ~1.772

v = gamma( 0.0 );
// returns Infinity

v = gamma( -0.0 );
// returns -Infinity

v = gamma( 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 gamma = require( '@stdlib/math-base-special-gamma' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 100, -10.0, 10.0, opts );

logEachMap( 'x: %0.4f, f(x): %0.4f', x, gamma );
```

</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/gamma.h"
```

#### stdlib_base_gamma( x )

Evaluates the [gamma function][gamma-function].

```c
double out = stdlib_base_gamma( 4.0 );
// returns 6.0

out = stdlib_base_gamma( -1.5 );
// returns ~2.363
```

The function accepts the following arguments:

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

```c
double stdlib_base_gamma( 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/gamma.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 4.0, -1.5, -0.5, 0.5 };

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

* * *

## See Also

-   <span class="package-name">[`@stdlib/math-base/special/gamma1pm1`][@stdlib/math/base/special/gamma1pm1]</span><span class="delimiter">: </span><span class="description">compute gamma(x+1) - 1.</span>
-   <span class="package-name">[`@stdlib/math-base/special/gammainc`][@stdlib/math/base/special/gammainc]</span><span class="delimiter">: </span><span class="description">incomplete gamma function.</span>
-   <span class="package-name">[`@stdlib/math-base/special/gammaincinv`][@stdlib/math/base/special/gammaincinv]</span><span class="delimiter">: </span><span class="description">inverse incomplete gamma function.</span>
-   <span class="package-name">[`@stdlib/math-base/special/gammaln`][@stdlib/math/base/special/gammaln]</span><span class="delimiter">: </span><span class="description">natural logarithm of the gamma function.</span>

</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

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

---

## Copyright

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

</section>

<!-- /.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-gamma/tree/deno
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[branches-url]: https://github.com/stdlib-js/math-base-special-gamma/blob/main/branches.md

[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function

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

[real]: https://en.wikipedia.org/wiki/Real_number

[complex]: https://en.wikipedia.org/wiki/Complex_number

[@stdlib/constants/float64/eulergamma]: https://www.npmjs.com/package/@stdlib/constants-float64-eulergamma

<!-- <related-links> -->

[@stdlib/math/base/special/gamma1pm1]: https://www.npmjs.com/package/@stdlib/math-base-special-gamma1pm1

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

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

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

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