npm package: @stdlib/math-base-special-betaln
Description: Natural logarithm of the beta function.
Author: The Stdlib Authors
Latest version: 0.3.1

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# betaln

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

> [Natural logarithm][natural-logarithm] of the [beta function][beta-function].

<section class="intro">

The [beta function][beta-function], also called the Euler integral, is defined as

<!-- <equation class="equation" label="eq:beta_function" align="center" raw="\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t" alt="Equation for the beta function."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t" data-equation="eq:beta_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/betaln/docs/img/equation_beta_function.svg" alt="Equation for the beta function.">
    <br>
</div> -->

<!-- </equation> -->

The [beta function][beta-function] is related to the [gamma function][gamma-function] via the following equation

<!-- <equation class="equation" label="eq:beta_function2" align="center" raw="\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!" alt="Beta function expressed in terms of the Gamma function."> -->

<!-- <div class="equation" align="center" data-raw-text="\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!" data-equation="eq:beta_function2">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/betaln/docs/img/equation_beta_function2.svg" alt="Beta function expressed in terms of the Gamma function.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="installation">

## Installation

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

</section>

<section class="usage">

## Usage

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

#### betaln( x, y )

Evaluates the [natural logarithm][natural-logarithm] of the [beta function][beta-function].

```javascript
var val = betaln( 0.0, 0.0 );
// returns Infinity

val = betaln( 1.0, 1.0 );
// returns 0.0

val = betaln( -1.0, 2.0 );
// returns NaN

val = betaln( 5.0, 0.2 );
// returns ~1.218

val = betaln( 4.0, 1.0 );
// returns ~-1.386
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

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

for ( x = 0; x < 10; x++ ) {
    for ( y = 10; y > 0; y-- ) {
        console.log( 'x: %d, \t y: %d, \t f(x,y): %d', x, y, betaln( x, y ) );
    }
}
```

</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/betaln.h"
```

#### stdlib_base_betaln( x, y )

Evaluates the [natural logarithm][natural-logarithm] of the [beta function][beta-function].

```c
double v = stdlib_base_betaln( 5.0, 0.2 );
// returns ~1.218
```

The function accepts the following arguments:

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

```c
double stdlib_base_betaln( const double x, const double y );
```

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

int main( void ) {
    const double x[] = { 24.0, 32.0, 48.0, 116.0, 33.0 };
    const double y[] = { 12.0, 6.0, 15.0, 52.0, 22.0 };

    double out;
    int i;
    for ( i = 0; i < 5; i++ ) {
        out = stdlib_base_betaln( x[ i ], y[ i ] );
        printf( "betaln(%lf, %lf) = %lf\n", x[ i ], y[ i ], out );
    }
}
```

</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/beta`][@stdlib/math/base/special/beta]</span><span class="delimiter">: </span><span class="description">beta function.</span>
-   <span class="package-name">[`@stdlib/math-base/special/betainc`][@stdlib/math/base/special/betainc]</span><span class="delimiter">: </span><span class="description">incomplete beta function.</span>
-   <span class="package-name">[`@stdlib/math-base/special/betaincinv`][@stdlib/math/base/special/betaincinv]</span><span class="delimiter">: </span><span class="description">inverse incomplete beta 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

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---

## Copyright

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

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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-betaln/tree/deno
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[natural-logarithm]: https://en.wikipedia.org/wiki/Natural_logarithm

[beta-function]: https://en.wikipedia.org/wiki/Beta_function

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

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

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

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

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

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</section>

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