@stdlib/math-base-special-gammainc/README.md

Incomplete gamma function.

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# gammainc

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Incomplete gamma function][incomplete-gamma-function].

<section class="intro">

Evaluates the regularized lower [incomplete gamma function][incomplete-gamma-function]:

<!-- <equation class="equation" label="eq:lower_incomplete_gamma" align="center" raw="P( x, a ) = \frac{\gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-t} \; dt" alt="Regularized lower incomplete gamma function."> -->

<div class="equation" align="center" data-raw-text="P( x, a ) = \frac{\gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-t} \; dt" data-equation="eq:lower_incomplete_gamma">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gammainc/docs/img/equation_lower_incomplete_gamma.svg" alt="Regularized lower incomplete gamma function.">
    <br>
</div> -->

<!-- </equation> -->

The function can also be used to evaluate the regularized upper incomplete gamma function, which is defined as follows:

<!-- <equation class="equation" label="eq:upper_incomplete_gamma" align="center" raw="Q( x, a ) = \frac{\Gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_x^\infty t^{a-1} e^{-t} \; dt" alt="Regularized upper incomplete gamma function."> -->

<!-- <div class="equation" align="center" data-raw-text="Q( x, a ) = \frac{\Gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_x^\infty t^{a-1} e^{-t} \; dt" data-equation="eq:upper_incomplete_gamma">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gammainc/docs/img/equation_upper_incomplete_gamma.svg" alt="Regularized upper incomplete gamma function.">
    <br>
</div> -->

<!-- </equation> -->

The two functions have the relationship `Q(x,a) = 1 - P(x,a)`.

In addition, this routine can be used to evaluate the _unregularized_ gamma functions. The range of above functions is `[0,1]`, which is not the case for the _unregularized_ versions. The unregularized lower incomplete gamma function is defined as

<!-- <equation class="equation" label="eq:unreg_lower_incomplete_gamma" align="center" raw="\gamma(a,x) = \int_0^x t^{a-1} e^{-t} \; dt" alt="Unregularized lower incomplete gamma function."> -->

<!-- <div class="equation" align="center" data-raw-text="\gamma(a,x) = \int_0^x t^{a-1} e^{-t} \; dt" data-equation="eq:unreg_lower_incomplete_gamma">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gammainc/docs/img/equation_unreg_lower_incomplete_gamma.svg" alt="Unregularized lower incomplete gamma function.">
    <br>
</div> -->

<!-- </equation> -->

and the upper unregularized incomplete gamma function is

<!-- <equation class="equation" label="eq:unreg_upper_incomplete_gamma" align="center" raw="\Gamma(a,x)= \int_x^\infty t^{a-1} e^{-t} \; dt" alt="Unregularized upper incomplete gamma function."> -->

<!-- <div class="equation" align="center" data-raw-text="\Gamma(a,x)= \int_x^\infty t^{a-1} e^{-t} \; dt" data-equation="eq:unreg_upper_incomplete_gamma">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gammainc/docs/img/equation_unreg_upper_incomplete_gamma.svg" alt="Unregularized upper incomplete gamma function.">
    <br>
</div>

<!-- </equation> -->

The relationship between the two functions is `Γ(a,x) = γ(a,x) + Γ(a)`.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/math-base-special-gammainc
```

</section>

<section class="usage">

## Usage

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

#### gammainc( x, s\[, regularized\[, upper ]] )

By default, evaluates the regularized lower [incomplete gamma function][incomplete-gamma-function] for inputs `x` and `s`. The third and fourth parameters of the function can be used to specify whether instead to evaluate the non-regularized and/or upper incomplete gamma functions, respectively.

```javascript
var y = gammainc( 6.0, 2.0 );
// returns ~0.9826

y = gammainc( 1.0, 2.0, true, true );
// returns ~0.7358

y = gammainc( 7.0, 5.0 );
// returns ~0.8270

y = gammainc( 7.0, 5.0, false );
// returns ~19.8482
```

If provided `NaN` as any argument, the function returns `NaN`.

```javascript
var y = gammainc( NaN, 1.0 );
// returns NaN

y = gammainc( 1.0, NaN );
// returns NaN
```

If provided a negative `x`, the function returns `NaN`.

```javascript
var y = gammainc( -2.0, 2.0 );
// returns NaN
```

If provided a nonpositive `s`, the function returns `NaN`.

```javascript
var y = gammainc( 2.0, -1.0 );
// returns NaN

y = gammainc( 2.0, 0.0 );
// returns NaN
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random-base-randu' );
var gammainc = require( '@stdlib/math-base-special-gammainc' );

var i;
var x;
var s;

for ( i = 0; i < 100; i++ ) {
    x = randu() * 10.0;
    s = randu() * 10.0;
    console.log( 'x: %d, \t s: %d, \t f(x,s): %d', x.toFixed( 4 ), s.toFixed( 4 ), gammainc( x, s ).toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<!-- 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/gamma1pm1`][@stdlib/math/base/special/gamma1pm1]</span><span class="delimiter">: </span><span class="description">compute gamma(x+1) - 1.</span>
-   <span class="package-name">[`@stdlib/math-base/special/gammaincinv`][@stdlib/math/base/special/gammaincinv]</span><span class="delimiter">: </span><span class="description">inverse incomplete gamma function.</span>
-   <span class="package-name">[`@stdlib/math-base/special/gammaln`][@stdlib/math/base/special/gammaln]</span><span class="delimiter">: </span><span class="description">natural logarithm 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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---

## Copyright

Copyright &copy; 2016-2024. The Stdlib [Authors][stdlib-authors].

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

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