time-to-botec/js/node_modules/@stdlib/math/base/special/betaln/README.md

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# betaln

> [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="usage">

## Usage

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

#### betaln( x, y )

Evaluates the 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 -->

<section class="links">

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

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

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

</section>

<!-- /.links -->