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MathsSpecialGamma

Stirling

viewed 1863 times and licensed 111 times
Stirling series approximation of the gamma function
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Info

Interface

C++

Stirling

 
doublestirlingdoublex )
This Stirling's series gives an approximate value for the factorial function x! or the gamma function \Gamma(x) 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:
There is an error with your graph parameters for stirling with options x=1:5 .log

Error Message:Function stirling failed. Ensure that: Invalid C++

See Also:

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

xa 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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