Regularized Gamma Function

The regularized gamma functions are defined by

$$P(a,z)=1-Q(a,z)\equiv {\gamma(a,z)\over\Gamma(a)}$$

and

$$Q(a,z)=1-P(a,z)\equiv {\Gamma(a,z)\over\Gamma(a)},$$

where $\gamma(a,z)$ and $\Gamma(a,z)$ are incomplete Gamma Functions and $\Gamma(a)$ is a complete Gamma Function. Their derivatives are

$${d\over dz} P(a,z) = e^{-z}z^{a-1}$$

$${d\over dz} Q(a,z) = -e^{-z}z^{a-1}.$$

See also Gamma Function, Regularized Beta Function

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 160-161, 1992.