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# factorialln

> Natural logarithm of the [factorial function][factorial-function].

<section class="intro">

The natural logarithm of the factorial function may be expressed

<!-- <equation class="equation" label="eq:factorialln_function" align="center" raw="f(n)=\ln (n!)" alt="Equation of the natural logarithm of the factorial."> -->

<div class="equation" align="center" data-raw-text="f(n)=\ln (n!)" data-equation="eq:factorialln_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/factorialln/docs/img/equation_factorialln_function.svg" alt="Equation of the natural logarithm of the factorial.">
    <br>
</div> -->

<!-- </equation> -->

The [factorial function][factorial-function] may be defined as the product

<!-- <equation class="equation" label="eq:factorial_function" align="center" raw="n! = \prod_{k=1}^n k" alt="Factorial function definition"> -->

<!-- <div class="equation" align="center" data-raw-text="n! = \prod_{k=1}^n k" data-equation="eq:factorial_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/factorialln/docs/img/equation_factorial_function.svg" alt="Factorial function definition">
    <br>
</div> -->

<!-- </equation> -->

or according to the recurrence relation

<!-- <equation class="equation" label="eq:factorial_recurrence_relation" align="center" raw="n! = \begin{cases}1 & \textrm{if } n = 0,\\(n-1)! \times n & \textrm{if } n > 1\end{cases}" alt="Factorial function recurrence relation"> -->

<!-- <div class="equation" align="center" data-raw-text="n! = \begin{cases}1 &amp; \textrm{if } n = 0,\\(n-1)! \times n &amp; \textrm{if } n &gt; 1\end{cases}" data-equation="eq:factorial_recurrence_relation">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/factorialln/docs/img/equation_factorial_recurrence_relation.svg" alt="Factorial function recurrence relation">
    <br>
</div> -->

<!-- </equation> -->

Following the convention for an [empty product][empty-product], in all definitions,

<!-- <equation class="equation" label="eq:zero_factorial" align="center" raw="0! = 1" alt="Zero factorial"> -->

<!-- <div class="equation" align="center" data-raw-text="0! = 1" data-equation="eq:zero_factorial">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/factorialln/docs/img/equation_zero_factorial.svg" alt="Zero factorial">
    <br>
</div> -->

<!-- </equation> -->

The [Gamma][gamma-function] function extends the [factorial function][factorial-function] for non-integer values.

<!-- <equation class="equation" label="eq:factorial_function_and_gamma" align="center" raw="n! = \Gamma(n+1)" alt="Factorial function extension via the Gamma function"> -->

<!-- <div class="equation" align="center" data-raw-text="n! = \Gamma(n+1)" data-equation="eq:factorial_function_and_gamma">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/factorialln/docs/img/equation_factorial_function_and_gamma.svg" alt="Factorial function extension via the Gamma function">
    <br>
</div>

<!-- </equation> -->

The [factorial][factorial-function] of a **negative** integer is not defined.

Evaluating the natural logarithm of [factorial function][factorial-function] is useful as the [factorial function][factorial-function] can overflow for large `n`. Thus, `factorialln( n )` is generally preferred to `ln( n! )`.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### factorialln( x )

Evaluates the natural logarithm of the [factorial function][factorial-function]. For input values other than negative integers, the function returns `ln( x! ) = ln( Γ(x+1) )`, where `Γ` is the [Gamma][gamma-function] function. For negative integers, the function returns `NaN`.

```javascript
var v = factorialln( 3.0 );
// returns ~1.792

v = factorialln( 2.4 );
// returns ~1.092

v = factorialln( -1.0 );
// returns NaN

v = factorialln( -1.5 );
// returns ~1.266
```

If provided `NaN`, the function returns `NaN`.

```javascript
var v = factorialln( NaN );
// returns NaN
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var incrspace = require( '@stdlib/array/base/incrspace' );
var factorialln = require( '@stdlib/math/base/special/factorialln' );

var x = incrspace( -10.0, 50.0, 0.5 );

var i;
for ( i = 0; i < x.length; i++ ) {
    console.log( 'x: %d, f(x): %d', x[ i ], factorialln( x[ i ] ) );
}
```

</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/factorialln.h"
```

#### stdlib_base_factorialln( x )

Evaluates the natural logarithm of the [factorial function][factorial-function]. For input values other than negative integers, the function returns `ln( x! ) = ln( Γ(x+1) )`, where `Γ` is the [Gamma][gamma-function] function. For negative integers, the function returns `NaN`.

```c
double out = stdlib_base_factorialln( 3.0 );
// returns ~1.792

out = stdlib_base_factorialln( -1.5 );
// returns ~1.266
```

The function accepts the following arguments:

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

```c
double stdlib_base_factorialln( 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/factorialln.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 2.0, 3.0, 5.0, 8.0 };

    double y;
    int i;
    for ( i = 0; i < 4; i++ ) {
        y = stdlib_base_factorialln( x[ i ] );
        printf( "factorialln(%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/factorial`][@stdlib/math/base/special/factorial]</span><span class="delimiter">: </span><span class="description">evaluate a factorial.</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">

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

[factorial-function]: https://en.wikipedia.org/wiki/Factorial

[empty-product]: https://en.wikipedia.org/wiki/Empty_product

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

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

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

</section>

<!-- /.links -->
