Incomplete gamma function

Summary

The incomplete gamma functions split the integral of the gamma function at a point $x$. The lower one $\gamma(s, x)$ integrates $t^{s-1} e^{-t}$ from $0$ to $x$, and the upper one $\Gamma(s, x)$ integrates it from $x$ to $\infty$. Their sum is $\Gamma(s)$. They give the distribution function of the gamma distribution and contain the exponential integral and the error function as special cases.