infgamma

The incomplete gamma function, often denoted as $\Gamma(s, x)$ or infgamma(s, x), is a special function in mathematics. It is defined as the integral of the gamma function from x to infinity: $\Gamma(s, x) = \int_x^\infty t^{s-1} e^{-t} dt$. Here, $s$ and $x$ are complex numbers, although it is frequently encountered with real arguments.

The incomplete gamma function is a generalization of the complete gamma function, $\Gamma(s) = \int_0^\infty t^{s-1} e^{-t} dt$. The complete gamma function is related to the factorial function for positive integers, where $\Gamma(n) = (n-1)!$. The incomplete gamma function, therefore, extends this concept to cases where the integration range is truncated.

There are two common forms of the incomplete gamma function: the upper incomplete gamma function, $\Gamma(s, x)$, and the lower incomplete gamma function, $\gamma(s, x)$. The lower incomplete gamma function is defined as $\gamma(s, x) = \int_0^x t^{s-1} e^{-t} dt$. The relationship between them is $\Gamma(s) = \gamma(s, x) + \Gamma(s, x)$.

The incomplete gamma function appears in various fields of mathematics and physics, including probability theory, statistics, and quantum mechanics. For example, it arises in the cumulative distribution functions of certain probability distributions, such as the gamma distribution and the Poisson distribution. It also plays a role in solving differential equations and in certain integral transforms. Computational algorithms exist for evaluating the incomplete gamma function for real and complex arguments.