CodeCogs

The regularized lower incomplete Gamma integral

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Interface

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

using namespace Maths::Special::Gamma;

double gammaLower_reg(double a, double x)

The regularized lower portion of the incomplete Gamma integral. This is also sometime referred to as the left tail of the integral, i.e

$\frac{\Gamma_{0,x}(a)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x e^{-t} t^{a-1}\ dt$

(1)

Accuracy:

Tested at random a, x.

Relative error:
domain     # trials      peak         rms
0,30       200000       3.6e-14     2.9e-15
0,100      300000       9.9e-14     1.5e-14

Example:

#include <stdio.h>
#include <codecogs/maths/special/gamma/gamma_lower_reg.h>
void main
{
  for(double a=0; a<3; a+=0.25)
  {
    double y=Maths::Special::Gamma::gammaLower_reg(a, 2);
    printf("\n gammaLower_reg(%lf, 2)=%lf", a,y);
  }
}

Output:

gammaLower_reg(0.000000, 2)=1.000000
gammaLower_reg(0.250000, 2)=0.982714
gammaLower_reg(0.500000, 2)=0.954500
gammaLower_reg(0.750000, 2)=0.914944
gammaLower_reg(1.000000, 2)=0.864665
gammaLower_reg(1.250000, 2)=0.805153
gammaLower_reg(1.500000, 2)=0.738536
gammaLower_reg(1.750000, 2)=0.667294
gammaLower_reg(2.000000, 2)=0.593994
gammaLower_reg(2.250000, 2)=0.521056
gammaLower_reg(2.500000, 2)=0.450584
gammaLower_reg(2.750000, 2)=0.384266

References

Cephes Math Library Release 2.8: June, 2000

Parameters

input argument

upper positive limit of the integral.

In this implementation, both parameters must be positive.

Interactive Calculator

Result –

Gamma_Lower_Reg