CodeCogs

A Sterling series approximation of the gamma function.

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Interface

#include <codecogs/maths/special/gamma/gamma_simple.h>

using namespace Maths::Special::Gamma;

inline double gamma_simple(double x)

Returns a simple approximation to the gamma functions. If your only interested in low levels of accuracy (10 significant figures), then this solution is evaluated quickly and is relatively stable.

This approximation of the gamma function is achieved using the exponential of the Stirling series,

$\Gamma(z) = e^{\ln |\Gamma(z)|}$

(1)

where

$ln \Gamma(z) = \frac{1}{2} \ln(2 \pi) + (z - \frac{1}{2}) \ln z - z + \frac{1}{12z} - \frac{1}{360 z^3} + \frac{1}{1260 z^5} - ...$

(2)

Example 1

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

int main()
{
  for(double x=3; x<7; x+=0.4)
  printf("\n x=%lf gamma_simple=%lf",x, Maths::Special::Gamma::gamma_simple(x));
  return 0;
}

Output:

x=3.000000 gamma_simple=2.000000
x=3.400000 gamma_simple=2.981206
x=3.800000 gamma_simple=4.694174
x=4.200000 gamma_simple=7.756690
x=4.600000 gamma_simple=13.381286
x=5.000000 gamma_simple=24.000000
x=5.400000 gamma_simple=44.598848
x=5.800000 gamma_simple=85.621738
x=6.200000 gamma_simple=169.406099
x=6.600000 gamma_simple=344.701924

Parameters

argument

Interactive Calculator

Result –

Gamma_Simple

Interface

Example 1

Parameters