Incomplete Gamma Function

[Dieter]

(03-26-2015 10:41 PM)bshoring Wrote:  Do you think you can shed any light on this for me? I would just like to be able to understand what I am getting.

There are different Gamma functions.

1. The "normal" Gamma function Γ(a). For integer a this is equal to (a–1)!. So Γ(5) = 4! = 24.
This, let's say "complete" Gamma function Γ(a) can be expressed as an integral from 0 to infinity.

2. The incomplete Gamma functions γ(a, x) and Γ(a, x). These are the integrals from 0 to x resp. from x to infinity. So they sum up to Γ(a).
Example: γ(5, 3) + Γ(5, 3) = 4,4337 + 19,5663 = 24 = Γ(5).
Since Γ(a) is the integral from 0 to infinity, and Γ(a, x) is the integral from x to infinity, it's clear that Γ(a, 0) = Γ(a).

3. The regularized Gamma functions P(a, x) and Q(a, x). These are simply γ(a, x) resp. Γ(a, x) divided by Γ(a). So they sum up to 1.
Example: P(5, 3) + Q(5, 3) = 4,4337/24 + 19,5663/24 = 0,18474 + 0,81526 = 1.

Dieter