@stdlib/math-base-special-gamma-lanczos-sum
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Calculate the Lanczos sum for the approximation of the gamma function.
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# Gamma Lanczos Sum
[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->
> Calculate the 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/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.
</section>
<!-- /.intro -->
<section class="installation">
## Installation
```bash
npm install @stdlib/math-base-special-gamma-lanczos-sum
```
</section>
<section class="usage">
## Usage
```javascript
var gammaLanczosSum = require( '@stdlib/math-base-special-gamma-lanczos-sum' );
```
#### gammaLanczosSum( x )
Calculates the Lanczos sum for the approximation of the [gamma function][gamma-function].
```javascript
var v = gammaLanczosSum( 4.0 );
// returns ~950.366
v = gammaLanczosSum( -1.5 );
// returns ~1373366.245
v = gammaLanczosSum( -0.5 );
// returns ~-699841.735
v = gammaLanczosSum( 0.5 );
// returns ~96074.186
v = gammaLanczosSum( 0.0 );
// returns Infinity
v = gammaLanczosSum( NaN );
// returns NaN
```
</section>
<!-- /.usage -->
<section class="examples">
## Examples
<!-- eslint no-undef: "error" -->
```javascript
var linspace = require( '@stdlib/array-base-linspace' );
var gammaLanczosSum = require( '@stdlib/math-base-special-gamma-lanczos-sum' );
var x = linspace( -10.0, 10.0, 100 );
var i;
for ( i = 0; i < x.length; i++ ) {
console.log( 'x: %d, f(x): %d', x[ i ], gammaLanczosSum( x[ i ] ) );
}
```
</section>
<!-- /.examples -->
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* * *
<section class="c">
## C APIs
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<section class="intro">
</section>
<!-- /.intro -->
<!-- C usage documentation. -->
<section class="usage">
### Usage
```c
#include "stdlib/math/base/special/gamma_lanczos_sum.h"
```
#### stdlib_base_gamma_lanczos_sum( x )
Calculates the Lanczos sum for the approximation of the [gamma function][gamma-function].
```c
double out = stdlib_base_gamma_lanczos_sum( 4.0 );
// returns ~950.366
out = stdlib_base_gamma_lanczos_sum( -1.5 );
// returns ~1373366.245
```
The function accepts the following arguments:
- **x**: `[in] double` input value.
```c
double stdlib_base_gamma_lanczos_sum( 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.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( x[ i ] );
printf( "gamma_lanczos_sum(%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-expg-scaled`][@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled]</span><span class="delimiter">: </span><span class="description">calculate a scaled Lanczos sum for the approximation of the gamma function.</span>
</section>
<!-- /.related -->
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<section class="main-repo" >
* * *
## Notice
This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.
For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].
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---
## License
See [LICENSE][stdlib-license].
## Copyright
Copyright © 2016-2024. The Stdlib [Authors][stdlib-authors].
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[@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
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[@stdlib/math/base/special/gamma]: https://www.npmjs.com/package/@stdlib/math-base-special-gamma
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