Incomplete Gamma Function

[bshoring]

OK, I'm starting to see it. It appears to me the Henrici program gives us the Γ(a, x). Given (5,3), it yields 19,5663, which, if added to y(5,3), or 4,4337, gives us 24.

Then, would the above program be called the upper incomplete gamma?

Thanks,
Bob

(03-26-2015 11:10 PM)Dieter Wrote:  

(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