Gamma Scaled Lanczos Sum

Calculate a scaled Lanczos sum for the approximation of the gamma function.

The Lanczos approximation for the gamma function can be written in partial fraction form as follows:

normal upper Gamma left-parenthesis n right-parenthesis equals StartFraction left-parenthesis n plus g minus 0.5 right-parenthesis Superscript n minus 0.5 Baseline Over e Superscript n plus g minus 0.5 Baseline EndFraction upper L Subscript g Baseline left-parenthesis n right-parenthesis

where g is an arbitrary constant and L_g(n) is the Lanczos sum. The scaled Lanczos sum is given by

upper L Subscript g Baseline left-parenthesis n right-parenthesis dot exp left-parenthesis negative g right-parenthesis

Usage

var gammaLanczosSumExpGScaled = require( '@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled' );

gammaLanczosSumExpGScaled( x )

Calculates the Lanczos sum for the approximation of the gamma function (scaled by exp(-g), where g = 10.900511).

var v = gammaLanczosSumExpGScaled( 4.0 );
// returns ~0.018

v = gammaLanczosSumExpGScaled( -1.5 );
// returns ~25.337

v = gammaLanczosSumExpGScaled( -0.5 );
// returns ~-12.911

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

v = gammaLanczosSumExpGScaled( 0.0 );
// returns Infinity

v = gammaLanczosSumExpGScaled( NaN );
// returns NaN

Examples

var linspace = require( '@stdlib/math/utils/linspace' );
var gammaLanczosSumExpGScaled = require( '@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled' );

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

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