 Incomplete gamma function

For real $a>0$ and $x\ge 0$, the lower and upper incomplete gamma functions are respectivelyThey divide the Gamma function integral at $x$, so $\gamma (a,x)+\Gamma (a,x)=\Gamma (a)$. The normalized ratios $P(a,x)=\gamma (a,x)/\Gamma (a)$ and $Q(a,x)=\Gamma (a,x)/\Gamma (a)$ sum to1. Lowercase $\gamma$ and uppercase $\Gamma$ distinguish the two tails; using one symbol for both would reverse which part is omitted.^[ref]

For example, $\gamma (1,x)=1-e^{-x}$ and $\Gamma (1,x)=e^{-x}$. Thus $P$ is the cumulative probability and $Q$ the tail probability for a gamma distribution of unit scale. Integration by parts gives $\Gamma (a+1,x)=a\Gamma (a,x)+x^a{e}^{-x}$, a recurrence useful for both numerical evaluation and endpoint asymptotic expansions.

The incomplete gamma function is a mathematical function that generalizes the gamma function, which itself is a fundamental function in mathematics, particularly in the fields of statistics and probability theory. The incomplete gamma function is useful in various applications, including statistical distributions and hypothesis testing. The incomplete gamma function is defined in two forms: the lower incomplete gamma function and the upper incomplete gamma function.