npm package: @stdlib/math-base-special-hyp2f1
Description: Evaluates the Gaussian hypergeometric function.
Author: The Stdlib Authors
Latest version: 0.1.1

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</details>

# Gaussian hypergeometric function

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

> 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="installation">

## Installation

```bash
npm install @stdlib/math-base-special-hyp2f1
```

</section>

<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 -->

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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].

#### Community

[![Chat][chat-image]][chat-url]

---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-special-hyp2f1/main/LICENSE

[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>

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