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Gaussian hypergeometric function

Evaluates the Gaussian hypergeometric function.

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

$${}_2F_1(a, b; c; x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{x^n}{n!} = 1 + \frac{a b}{c} x + \frac{a(a+1) b(b+1)}{c(c+1)} \frac{x^2}{2!} + \frac{a(a+1)(a+2) b(b+1)(b+2)}{c(c+1)(c+2)} \frac{x^3}{3!} + \cdots$$

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

Here (q)ₙ is the (rising) Pochhammer symbol, which is defined by:

$$(q)_n = \begin{cases} 1 & n = 0 \\ q(q+1) \cdots (q+n-1) & n > 0 \end{cases}$$

For |x| >= 1, the function can be analytically continued using functional identities and transformation formulas.

Installation

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 available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md 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

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

hyp2f1( a, b, c, x )

Evaluates the Gaussian hypergeometric function.

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

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 );

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C APIs

Usage

#include "stdlib/math/base/special/hyp2f1.h"

stdlib_base_hyp2f1( a, b, c, x )

Evaluates the Gaussian hypergeometric function.

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.

double stdlib_base_hyp2f1( const double a, const double b, const double c, const double x );

Examples

#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 );
    }
}

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Notice

This package is part of 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, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2025. The Stdlib Authors.