Incomplete Gamma Function

The lower incomplete gamma function, \(\gamma\), is defined by the integral
\[ \gamma(s, x) = \int_0^x t^{s-1} e^{-t} \mathrm{d}t \]
Similarly, the upper incomplete gamma function, \(\Gamma\), is defined by the integral
\[ \gamma(s, x) = \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t \]
These are related to the gamma function by
\[ \gamma(s, x) + \Gamma(s, x) = \Gamma(s) \]