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# Gamma Scaled Lanczos Sum

> Calculate a scaled Lanczos sum for the approximation of the [gamma function][gamma-function].

<section class="intro">

The [Lanczos approximation][lanczos-approximation] for the [gamma function][gamma-function] can be written in partial fraction form as follows:

<!-- <equation class="equation" label="eq:lanczos_approximation" align="center" raw="\Gamma ( n ) = \frac{(n+g-0.5)^{n-0.5}}{e^{n+g-0.5}} L_g(n)" alt="Lanczos approximation for gamma function."> -->

<div class="equation" align="center" data-raw-text="\Gamma ( n ) = \frac{(n+g-0.5)^{n-0.5}}{e^{n+g-0.5}} L_g(n)" data-equation="eq:lanczos_approximation">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled/docs/img/equation_lanczos_approximation.svg" alt="Lanczos approximation for gamma function.">
    <br>
</div> -->

<!-- </equation> -->

where `g` is an [arbitrary constant][@stdlib/constants/float64/gamma-lanczos-g] and `L_g(n)` is the Lanczos sum. The scaled Lanczos sum is given by

<!-- <equation class="equation" label="eq:scaled_lanczos_sum" align="center" raw="L_g(n) \cdot \exp(-g)" alt="Scaled Lanczos sum."> -->

<!-- <div class="equation" align="center" data-raw-text="L_g(n) \cdot \exp(-g)" data-equation="eq:scaled_lanczos_sum">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled/docs/img/equation_scaled_lanczos_sum.svg" alt="Scaled Lanczos sum.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var gammaLanczosSumExpGScaled = require( '@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled' );
```

#### gammaLanczosSumExpGScaled( x )

Calculates the Lanczos sum for the approximation of the [gamma function][gamma-function] (scaled by `exp(-g)`, where `g = 10.900511`).

```javascript
var v = gammaLanczosSumExpGScaled( 4.0 );
// returns ~0.018

v = gammaLanczosSumExpGScaled( -1.5 );
// returns ~25.337

v = gammaLanczosSumExpGScaled( -0.5 );
// returns ~-12.911

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

v = gammaLanczosSumExpGScaled( 0.0 );
// returns Infinity

v = gammaLanczosSumExpGScaled( 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 gammaLanczosSumExpGScaled = require( '@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled' );

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

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

</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_lanczos_sum_expg_scaled.h"
```

#### stdlib_base_gamma_lanczos_sum_expg_scaled( x )

Calculates the Lanczos sum for the approximation of the [gamma function][gamma-function] (scaled by `exp(-g)`, where `g = 10.900511`).

```c
double out = stdlib_base_gamma_lanczos_sum_expg_scaled( 4.0 );
// returns ~0.018

out = stdlib_base_gamma_lanczos_sum_expg_scaled( -1.5 );
// returns ~25.337
```

The function accepts the following arguments:

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

```c
double stdlib_base_gamma_lanczos_sum_expg_scaled( 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_lanczos_sum_expg_scaled.h"
#include <stdlib.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_lanczos_sum_expg_scaled( x[ i ] );
        printf( "gamma_lanczos_sum_expg_scaled(%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/gamma`][@stdlib/math/base/special/gamma]</span><span class="delimiter">: </span><span class="description">gamma function.</span>
-   <span class="package-name">[`@stdlib/math/base/special/gamma-lanczos-sum`][@stdlib/math/base/special/gamma-lanczos-sum]</span><span class="delimiter">: </span><span class="description">calculate the Lanczos sum for the approximation of the gamma 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">

[@stdlib/constants/float64/gamma-lanczos-g]: https://www.npmjs.com/package/@stdlib/constants-float64-gamma-lanczos-g

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

[lanczos-approximation]: https://en.wikipedia.org/wiki/Lanczos_approximation

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

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

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

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

</section>

<!-- /.links -->
