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# gammainc

> [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="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 uniform = require( '@stdlib/random/array/uniform' );
var logEachMap = require( '@stdlib/console/log-each-map' );
var gammainc = require( '@stdlib/math/base/special/gammainc' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 100, 0.0, 10.0, opts );
var s = uniform( 100, 0.0, 10.0, opts );

logEachMap( 'x: %0.4f, \t s: %0.4f, \t f(x,s): %0.4f', x, s, gammainc );
```

</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/gammainc.h"
```

#### stdlib_base_gammainc( x, a, regularized, upper )

Evaluates the [incomplete gamma function][incomplete-gamma-function] for inputs `x` and `a`.

```c
double out = stdlib_base_gammainc( 0.0, 1.0, true, false );
// returns 0.0

out = stdlib_base_gammainc( 6.0, 2.0, true, false );
// returns ~0.9826
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.
-   **a**: `[in] double` input value.
-   **regularized**: `[in] bool` boolean indicating whether to evaluate a regularized incomplete gamma function.
-   **upper**: `[in] bool` boolean indicating whether to evaluate the upper incomplete gamma function.

```c
double stdlib_base_gammainc( const double x, const double a, const bool regularized, const bool upper );
```

</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/gammainc.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double x;
    double a;
    double y;
    int i;

    for ( i = 0; i < 10; i++ ) {
        x = random_uniform( 0.0, 1000.0 );
        a = random_uniform( 1.0, 1000.0 );
        y = stdlib_base_gammainc( x, a, true, false );
        printf( "x: %lf, s: %lf, f(x,s): %lf\n", x, a, 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/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 -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

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

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

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

[@stdlib/math/base/special/gamma1pm1]: https://github.com/stdlib-js/math/tree/main/base/special/gamma1pm1

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

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

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

</section>

<!-- /.links -->
