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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 [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 -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

[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://github.com/stdlib-js/math/tree/main/base/special/beta

[@stdlib/math/base/special/betainc]: https://github.com/stdlib-js/math/tree/main/base/special/betainc

[@stdlib/math/base/special/betaincinv]: https://github.com/stdlib-js/math/tree/main/base/special/betaincinv

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

</section>

<!-- /.links -->
