https://github.com/stdlib-js/math-base-special-hyp2f1
Evaluates the Gaussian hypergeometric function.
https://github.com/stdlib-js/math-base-special-hyp2f1
function gaussian hyp2f1 hypergeometric hypergeometric-function javascript math mathematics node node-js nodejs number special special-function special-functions stdlib
Last synced: 10 months ago
JSON representation
Evaluates the Gaussian hypergeometric function.
- Host: GitHub
- URL: https://github.com/stdlib-js/math-base-special-hyp2f1
- Owner: stdlib-js
- License: apache-2.0
- Created: 2025-04-01T06:06:33.000Z (11 months ago)
- Default Branch: main
- Last Pushed: 2025-04-07T07:12:50.000Z (10 months ago)
- Last Synced: 2025-04-07T07:34:54.902Z (10 months ago)
- Topics: function, gaussian, hyp2f1, hypergeometric, hypergeometric-function, javascript, math, mathematics, node, node-js, nodejs, number, special, special-function, special-functions, stdlib
- Language: JavaScript
- Homepage: https://github.com/stdlib-js/stdlib
- Size: 400 KB
- Stars: 0
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- Changelog: CHANGELOG.md
- Contributing: CONTRIBUTING.md
- License: LICENSE
- Code of conduct: CODE_OF_CONDUCT.md
- Citation: CITATION.cff
- Security: SECURITY.md
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README
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# Gaussian hypergeometric function
[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url]
> Evaluates the [Gaussian hypergeometric function][hypergeometric-function].
The [Gaussian hypergeometric function][hypergeometric-function] is defined for `|x| < 1` by the power series:
```math
{}_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
```
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:
```math
(q)_n = \begin{cases} 1 & n = 0 \\ q(q+1) \cdots (q+n-1) & n > 0 \end{cases}
```
For `|x| >= 1`, the function can be [analytically continued][analytic-continuation] using functional identities and transformation formulas.
## Installation
```bash
npm install @stdlib/math-base-special-hyp2f1
```
Alternatively,
- To load the package in a website via a `script` tag without installation and bundlers, use the [ES Module][es-module] available on the [`esm`][esm-url] branch (see [README][esm-readme]).
- If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions).
- For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [`umd`][umd-url] branch (see [README][umd-readme]).
The [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships.
To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.
## 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
```
## Examples
```javascript
var linspace = require( '@stdlib/array-base-linspace' );
var hyp2f1 = require( '@stdlib/math-base-special-hyp2f1' );
var a = linspace( -50.0, 50.0, 100 );
var b = linspace( -50.0, 50.0, 100 );
var c = linspace( -50.0, 50.0, 100 );
var x = linspace( -50.0, 50.0, 100 );
var i;
for ( i = 0; i < x.length; i++ ) {
console.log( 'a: %d, b: %d, c: %d, x: %d, 2F1(a,b;c;x): %d', a[ i ], b[ i ], c[ i ], x[ i ], hyp2f1( a[ i ], b[ i ], c[ i ], x[ i ] ) );
}
```
* * *
## C APIs
### 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 );
```
### Examples
```c
#include "stdlib/math/base/special/hyp2f1.h"
#include
#include
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 );
}
}
```
* * *
## 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 © 2016-2025. The Stdlib [Authors][stdlib-authors].
[npm-image]: http://img.shields.io/npm/v/@stdlib/math-base-special-hyp2f1.svg
[npm-url]: https://npmjs.org/package/@stdlib/math-base-special-hyp2f1
[test-image]: https://github.com/stdlib-js/math-base-special-hyp2f1/actions/workflows/test.yml/badge.svg?branch=main
[test-url]: https://github.com/stdlib-js/math-base-special-hyp2f1/actions/workflows/test.yml?query=branch:main
[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/math-base-special-hyp2f1/main.svg
[coverage-url]: https://codecov.io/github/stdlib-js/math-base-special-hyp2f1?branch=main
[chat-image]: https://img.shields.io/gitter/room/stdlib-js/stdlib.svg
[chat-url]: https://app.gitter.im/#/room/#stdlib-js_stdlib:gitter.im
[stdlib]: https://github.com/stdlib-js/stdlib
[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors
[umd]: https://github.com/umdjs/umd
[es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules
[deno-url]: https://github.com/stdlib-js/math-base-special-hyp2f1/tree/deno
[deno-readme]: https://github.com/stdlib-js/math-base-special-hyp2f1/blob/deno/README.md
[umd-url]: https://github.com/stdlib-js/math-base-special-hyp2f1/tree/umd
[umd-readme]: https://github.com/stdlib-js/math-base-special-hyp2f1/blob/umd/README.md
[esm-url]: https://github.com/stdlib-js/math-base-special-hyp2f1/tree/esm
[esm-readme]: https://github.com/stdlib-js/math-base-special-hyp2f1/blob/esm/README.md
[branches-url]: https://github.com/stdlib-js/math-base-special-hyp2f1/blob/main/branches.md
[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