igam

sherpa.utils.igam(a, x)

Calculate the regularized incomplete Gamma function (lower).

The function is defined using the complete Gamma function - gamma(a) - as:

igam(a,x) = 1/gamma(a) Int_0^x e^(-t) t^(a^-1) dt
Parameters:
  • a (scalar or array) – a > 0
  • x (scalar or array) – x > 0
Returns:

val
Return type:

scalar or array

See also

gamma()
The Gamma function.
igamc()
The complement of the incomplete Gamma function (upper).

Notes

In this implementation, which is provided by the Cephes Math Library [1], both arguments must be positive. The integral is evaluated by either a power series or continued fraction expansion, depending on the relative values of a and x. Using IEEE arithmetic, the relative errors are

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

References

[1]Cephes Math Library Release 2.0: April, 1987. Copyright 1985, 1987 by Stephen L. Moshier. Direct inquiries to 30 Frost Street, Cambridge, MA 02140.

Examples

>>> igam(1, 2)
0.8646647167633873

>>> igam([1,1], [2,3])
array([ 0.86466472,  0.95021293])