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

var x = linspace( -10.0, 10.0, 100 );
var v;
var i;

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

</section>

<!-- /.examples -->

<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

</section>

<!-- /.links -->