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# log gamma stirling

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Natural Logarithm of Gamma function
Controller: CodeCogs

C++

## Log Gamma Stirling

 doublelog_gamma_stirling( double x int* sign = NULL )
The logarithm of the gamma function is sometimes treated as a special function to avoid additional 'branch cut' stuctures that are introduced by the logarithm function (see http://mathworld.wolfram.com/LogGammaFunction.html )

The log-gamma function can be defined by
$\ln&space;\Gamma(z)&space;=&space;-\gamma&space;z&space;-&space;\ln&space;z&space;+&space;\sum_{k=1}^{\infty}&space;\left&space;[&space;\frac{z}{k}&space;-&space;\ln&space;\left&space;(&space;1&space;+&space;\frac{z}{k}&space;\right&space;)&space;\right&space;]$

Stirling series (in its normal form) also provides a solution, given by a simple analytic expression
$ln&space;\Gamma(z)&space;=&space;\frac{1}{2}&space;\ln(2&space;\pi)&space;+&space;(z&space;-&space;\frac{1}{2})&space;\ln&space;z&space;-&space;z&space;+&space;\frac{1}{12z}&space;-&space;\frac{1}{360&space;z^3}&space;+&space;\frac{1}{1260&space;z^5}&space;-&space;...$

The function returns the base e (2.718...) logarithm of the absolute value of the gamma function of the argument.

The sign (+1 or -1) of the gamma function is returned in a global (extern) variable named sgngam.

For arguments greater than 13, the logarithm of the gamma function is approximated by the logarithmic version of Stirling's formula using a polynomial approximation of degree 4. Arguments between -33 and +33 are reduced by recurrence to the interval [2,3] of a rational approximation.

The cosecant reflection formula is employed for arguments less than -33.

Arguments greater than MAXLGM return MAXNUM and an error message. MAXLGM = 2.035093e36 for DEC arithmetic or 2.556348e305 for IEEE arithmetic.

## Accuracy:

<pre> domain # trials peak rms 0, 3 28000 5.4e-16 1.1e-16 2.718, 2.556e305 40000 3.5e-16 8.3e-17 </pre> The error criterion was relative when the function magnitude was greater than one but absolute when it was less than one.

## Example:

#include <codecogs/maths/special/gamma/log_gamma_stirling.h>
#include <stdio.h>

int main()
{
for(double x=10; x<=20; x+=0.5)
printf("\n x=%lf log_gamma_stirling(x)=%lf",x, Maths::Special::Gamma::log_gamma_stirling(x));
}

## Output:

x=10.000000 log_gamma_stirling(x)=12.801827
x=10.500000 log_gamma_stirling(x)=13.940625
x=11.000000 log_gamma_stirling(x)=15.104413
x=11.500000 log_gamma_stirling(x)=16.292000
x=12.000000 log_gamma_stirling(x)=17.502308
x=12.500000 log_gamma_stirling(x)=18.734348
x=13.000000 log_gamma_stirling(x)=19.987214
x=13.500000 log_gamma_stirling(x)=21.260076
x=14.000000 log_gamma_stirling(x)=22.552164
x=14.500000 log_gamma_stirling(x)=23.862766
x=15.000000 log_gamma_stirling(x)=25.191221
x=15.500000 log_gamma_stirling(x)=26.536914
x=16.000000 log_gamma_stirling(x)=27.899271
x=16.500000 log_gamma_stirling(x)=29.277755
x=17.000000 log_gamma_stirling(x)=30.671860
x=17.500000 log_gamma_stirling(x)=32.081115
x=18.000000 log_gamma_stirling(x)=33.505073
x=18.500000 log_gamma_stirling(x)=34.943316
x=19.000000 log_gamma_stirling(x)=36.395445
x=19.500000 log_gamma_stirling(x)=37.861087
x=20.000000 log_gamma_stirling(x)=39.339884

## References:

• Cephes Math Library Release 2.8: June, 2000
• http://mathworld.wolfram.com/LogGammaFunction.html

### Parameters

 x is primary argument. sign is a pointer to a variable, into which the sign of the solution is placed, if sign not equal to NULL (default=NULL).

### Authors

Stephen L.Moshier. Copyright 1984, 1987, 1989, 1992, 2000
Updated by Will Bateman
##### Source Code

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Last Modified: 18 Oct 07 @ 17:07     Page Rendered: 2022-03-14 14:59:48