Incomplete gamma function is defined as

These formulas define the lower incomplete gamma function and the upper incomplete gamma function (depending on whether the lower or upper limit of integration is fixed). The regularized gamma function is closely related to incomplete gamma functions:

Incomplete gamma functions are often met in statistics. For example, the cumulative distribution functions of gamma distribution, Poisson distribution and chi-square distribution could be written by using the incomplete gamma function.

**IncompleteGamma** subroutine calculates *P(a,x)*, **IncompleteGammaC** subroutine calculates *Q(a,x)*. Inverse *Q(a,x)* function is calculated by using the **InvIncompleteGammaC** subroutine. To calculate the values, the algorithm uses series expansion or continued fractions representation (depending on an argument value).

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