Gamma Function

Gamma function.

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

normal upper Gamma left-parenthesis n right-parenthesis equals left-parenthesis n minus 1 right-parenthesis factorial

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

normal upper Gamma left-parenthesis z right-parenthesis equals StartFraction e Superscript minus gamma z Baseline Over z EndFraction product Underscript n equals 1 Overscript normal infinity Endscripts left-parenthesis 1 plus StartFraction z Over n EndFraction right-parenthesis Superscript negative 1 Baseline e Superscript z slash n

where γ ≈ 0.577216 is the Euler–Mascheroni constant.

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

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

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