# Integartion of Incomplete Gamma Function

1. Sep 18, 2008

### bill august

Hi all,
Recently I am struglling with the nested integration of incomplete gamma function.
$$\int_{0}^{\infty}\int_{0}^{y}x^{\alpha-1}e^{-x/\beta}y^{\alpha-1}e^{-y/\beta}dx dy$$
after integre 'x', we can get
$$\int_{0}^{\infty}\beta^{\alpha}\gamma(\alpha, y/\beta)y^{\alpha-1}e^{-y/\beta }dy$$
I know that i can integre it directly and get the result which involves gussian hypergeometric function 2F1. But i want to use the series expansion,
$$\gamma(\alpha, x) = x^\alpha\sum_{k=0}^{\infty}\frac{(-x)^k}{(\alpha+k)k!}$$
The result is,
$$\sum_{k=0}^{\infty}\frac{(-1)^k\beta^{2\alpha}\Gamma(2\alpha+k)}{(\alpha+k)k!}$$
The problem is that it do not always converge, any idea?
Many thanks indeed.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted