The Incomplete Gamma Function

This entry provides two special functions, the lower and upper incomplete gamma functions. Similarly to the ‘complete’ gamma function $\Gamma(s)$, which is defined as $\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}\,\text{d}t$ for $\text{Re}(s)>0$ and by analytic continuation elsewhere, these are defined as $\gamma(s,z) = \int_0^z t^{s-1}e^{-t}\,\text{d}t$ and $\Gamma(s,z) = \int_z^\infty t^{s-1}e^{-t}\,\text{d}t$, respectively, for $\text{Re}(s)>0$ and analytically continued to the entire complex plane.

$\gamma(s,z)$ is constructed using the regularised hypergeometric series and $\Gamma(s,z)$ via its contour integral representation. Various results are provided, including:

• holomorphicity, continuity, limits, and derivatives
• series and integral representations
• shift identities such as $\Gamma(s+1, z) = s\Gamma(s,z) + z^s e^{-z}$
• the identity $\gamma(s,z) + \Gamma(s,z) = \Gamma(s)$
• the fact that $\Gamma(s,z) \to \Gamma(s)$ as $z\to 0$ within a certain region
• closed forms for $\Gamma(n,z)$ and $\gamma(n,z)$, where $n$ is a positive integer
• the connection to the error function via $\Gamma(\frac{1}{2}, z) = \sqrt{\pi}\cdot\text{erf}(\sqrt{z})$