The following page includes the definitions of the gamma functions and their relations to each other. These functions were particularly useful in the development of promethium.

Gamma Function

$\displaystyle \Gamma(z) = \int_{0}^{\infty} t^{z - 1} e^{-t} dt$

Incomplete Gamma Function

Lower Incomplete Gamma Function

$\displaystyle \gamma(s, x) = \int_{0}^{x} t^{s - 1} e^{-t} dt$

Upper Incomplete Gamma Function

$\displaystyle \Gamma(s, x) = \int_{x}^{\infty} t^{s - 1} e^{-t} dt$

Regularized Incomplete Gamma Function

Regularized Lower Incomplete Gamma Function

$\displaystyle P(a, x) = \frac{1}{\Gamma(a)} \int_{0}^{x} t^{a - 1} e^{-t} dt$

Regularized Upper Incomplete Gamma Function

$\displaystyle Q(a, x) = \frac{1}{\Gamma(a)} \int_{x}^{\infty} t^{a - 1} e^{-t} dt$

.

$P(a, x) + Q(a, x) = 1$