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# Stirling

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Stirling series approximation of the gamma function
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## Stirling

 doublestirling( double x )
This Stirling's series gives an approximate value for the factorial function x! or the gamma function  for x>>1.

The asymptotic series for a gamma function is given by


The polynomial STIR within stirling are only valid for 33 <= x <= 172. Please note: The upper limit will depend on the accuracy of your computer.

Graphically this function has the form:

Maths/Special/Gamma/Gamma and Maths/Special/Gamma/Gamma_Simple

## Accuracy:

In comparison to integer fractorial, this approximate it accurate to about 15dp at x=35, but obviously very poor for lower values of x (i.e. x<6)

### Example 1

Compared the outut from Stirling with a traditional factorial
#include <codecogs/maths/special/gamma/stirling.h>
#include <stdio.h>

int main ()
{
for(double x=33; x<=40; x+=0.5)
printf("\n x=%lf stirling(x)=%.0lf",x, Maths::Special::Gamma::stirling(x));
}

## Output:

x=33.000000 stirling(x)=263130836933693517766352317727113216
x=33.500000 stirling(x)=1505856975626702287543640208112091136
x=34.000000 stirling(x)=8683317618811887119307294612729626624
x=34.500000 stirling(x)=50446208683494518220996649360199516160
x=35.000000 stirling(x)=295232799039604195113013396920323801088
x=35.500000 stirling(x)=1740394199580561001405912957537658863616
x=36.000000 stirling(x)=10333147966386146640060809577425524490240
x=36.500000 stirling(x)=61783994085109902705073076587151908405248
x=37.000000 stirling(x)=371993326789901177492420297158468206329856
x=37.500000 stirling(x)=2255115784106511937141198419741231238086656
x=38.000000 stirling(x)=13763753091226345578872114833606270345281536
x=38.500000 stirling(x)=84566841903994165920581434052426627138191360
x=39.000000 stirling(x)=523022617466601117141859892252474974331207680
x=39.500000 stirling(x)=3255823413303776398101457267888729462505734144
x=40.000000 stirling(x)=20397882081197441587828472941238084160318341120

## References:

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

### Parameters

 x a value.

### Returns

an approximation of the factorial of x.

### Authors

Stephen L.Moshier. Copyright 1984, 1987, 1989, 1992, 2000
Documentation by Will Bateman (August 2005)
##### Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

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