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Gamma Function

Gamma function.

The gamma function extends the factorial function to real and complex numbers. If n is a positive integer,

$$\Gamma ( n ) = (n-1)!$$

Generalized to all complex numbers z, except for nonpositive integers, the gamma function can be expressed as an infinite product

$$\Gamma ( z ) = \frac{e^{-\gamma z}}{z} \prod^{\infty}_{n=1} \left ( 1+\frac{z}{n}\right )^{-1} e^{z/n}$$

where γ ≈ 0.577216 is the Euler–Mascheroni constant.

Installation

npm install @stdlib/math-base-special-gamma

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

gamma( x )

Evaluates the gamma function.

var v = gamma( 4.0 );
// returns 6.0

v = gamma( -1.5 );
// returns ~2.363

v = gamma( -0.5 );
// returns ~-3.545

v = gamma( 0.5 );
// returns ~1.772

v = gamma( 0.0 );
// returns Infinity

v = gamma( -0.0 );
// returns -Infinity

v = gamma( NaN );
// returns NaN

Examples

var linspace = require( '@stdlib/array-base-linspace' );
var gamma = require( '@stdlib/math-base-special-gamma' );

var x = linspace( -10.0, 10.0, 100 );

var i;
for ( i = 0; i < x.length; i++ ) {
    console.log( 'x: %d, f(x): %d', x[ i ], gamma( x[ i ] ) );
}

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C APIs

Usage

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

stdlib_base_gamma( x )

Evaluates the gamma function.

double out = stdlib_base_gamma( 4.0 );
// returns 6.0

out = stdlib_base_gamma( -1.5 );
// returns ~2.363

The function accepts the following arguments:

• x: [in] double input value.

double stdlib_base_gamma( const double x );

Examples

#include "stdlib/math/base/special/gamma.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 4.0, -1.5, -0.5, 0.5 };

    double y;
    int i;
    for ( i = 0; i < 4; i++ ) {
        y = stdlib_base_gamma( x[ i ] );
        printf( "gamma(%lf) = %lf\n", x[ i ], y );
    }
}

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See Also

• @stdlib/math-base/special/gamma1pm1: compute gamma(x+1) - 1.
• @stdlib/math-base/special/gammainc: incomplete gamma function.
• @stdlib/math-base/special/gammaincinv: inverse incomplete gamma function.
• @stdlib/math-base/special/gammaln: natural logarithm of the gamma function.

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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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Copyright

Copyright © 2016-2024. The Stdlib Authors.