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# Gaussian hypergeometric function

> Evaluates the [Gaussian hypergeometric function][hypergeometric-function].

<section class="intro">

The [Gaussian hypergeometric function][hypergeometric-function] is defined for `|x| < 1` by the power series:

<!-- <equation class="equation" label="eq:hypergeometric_function" align="center" raw="{}_2F_1(a, b; c; x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{x^n}{n!} = 1 + \frac{a b}{c} x + \frac{a(a+1) b(b+1)}{c(c+1)} \frac{x^2}{2!} + \frac{a(a+1)(a+2) b(b+1)(b+2)}{c(c+1)(c+2)} \frac{x^3}{3!} + \cdots" alt="Gaussian hypergeometric function."> -->

<div class="equation" align="center" data-raw-text="{}_2F_1(a, b; c; x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{x^n}{n!} = 1 + \frac{a b}{c} x + \frac{a(a+1) b(b+1)}{c(c+1)} \frac{x^2}{2!} + \frac{a(a+1)(a+2) b(b+1)(b+2)}{c(c+1)(c+2)} \frac{x^3}{3!} + \cdots" data-equation="eq:hypergeometric_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/hyp2f1/docs/img/equation_hypergeometric_function.svg" alt="Gaussian hypergeometric function.">
    <br>
</div> -->

<!-- </equation> -->

and is undefined (or infinite) if `c` equals a non-positive integer.

Here `(q)ₙ` is the (rising) [Pochhammer symbol][pochhammer-symbol], which is defined by:

<!-- <equation class="equation" label="eq:pochhammer_symbol" align="center" raw="(q)_n = \begin{cases} 1 & n = 0 \\ q(q+1) \cdots (q+n-1) & n > 0 \end{cases}" alt="Pochhammer symbol."> -->

<!-- <div class="equation" align="center" data-raw-text="(q)_n = \begin{cases} 1 & n = 0 \\ q(q+1) \cdots (q+n-1) & n > 0 \end{cases}" data-equation="eq:pochhammer_symbol">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/hyp2f1/docs/img/equation_pochhammer_symbol.svg" alt="Pochhammer symbol.">
    <br>
</div>

<!-- </equation> -->

For `|x| >= 1`, the function can be [analytically continued][analytic-continuation] using functional identities and transformation formulas.

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var hyp2f1 = require( '@stdlib/math/base/special/hyp2f1' );
```

#### hyp2f1( a, b, c, x )

Evaluates the [Gaussian hypergeometric function][hypergeometric-function].

```javascript
var v = hyp2f1( 1.0, 1.0, 1.0, 0.0 );
// returns 1.0

v = hyp2f1( 10.0, 7.4, -1.8, -0.99 );
// returns ~0.423

v = hyp2f1( 3.0, 4.0, 7.0, 1.0 );
// returns +Infinity

v = hyp2f1( NaN, 3.0, 2.0, 0.5 );
// 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 hyp2f1 = require( '@stdlib/math/base/special/hyp2f1' );

var opts = {
    'dtype': 'float64'
};
var a = uniform( 100, -50.0, 50.0, opts );
var b = uniform( 100, -50.0, 50.0, opts );
var c = uniform( 100, -50.0, 50.0, opts );
var x = uniform( 100, -50.0, 50.0, opts );

logEachMap( 'a: %0.4f, b: %0.4f, c: %0.4f, x: %0.4f, 2F1(a,b;c;x): %0.4f', a, b, c, x, hyp2f1 );
```

</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/hyp2f1.h"
```

#### stdlib_base_hyp2f1( a, b, c, x )

Evaluates the [Gaussian hypergeometric function][hypergeometric-function].

```c
double out = stdlib_base_hyp2f1( 1.0, 1.0, 1.0, 0.0 );
// returns 1.0

out = stdlib_base_hyp2f1( 10.0, 7.4, -1.8, -0.99 );
// returns ~0.423
```

The function accepts the following arguments:

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

```c
double stdlib_base_hyp2f1( const double a, const double b, const double c, 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/hyp2f1.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 a;
    double b;
    double c;
    double x;
    double y;
    int i;

    for ( i = 0; i < 10; i++ ) {
        a = random_uniform( -50.0, 50.0 );
        b = random_uniform( -50.0, 50.0 );
        c = random_uniform( -50.0, 50.0 );
        x = random_uniform( -50.0, 50.0 );
        y = stdlib_base_hyp2f1( a, b, c, x );
        printf( "a: %lf, b: %lf, c: %lf, x: %lf, 2F1(a,b;c;x): %lf\n", a, b, c, x, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

</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">

[hypergeometric-function]: https://en.wikipedia.org/wiki/Hypergeometric_function

[pochhammer-symbol]: https://en.wikipedia.org/wiki/Falling_and_rising_factorials

[analytic-continuation]: https://en.wikipedia.org/wiki/Analytic_continuation

</section>

<!-- /.links -->
