Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

Recently I am struglling with the nested integration of incomplete gamma function.

[tex]\int_{0}^{\infty}\int_{0}^{y}x^{\alpha-1}e^{-x/\beta}y^{\alpha-1}e^{-y/\beta}dx dy[/tex]

after integre 'x', we can get

[tex]\int_{0}^{\infty}\beta^{\alpha}\gamma(\alpha, y/\beta)y^{\alpha-1}e^{-y/\beta }dy[/tex]

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,

[tex]\gamma(\alpha, x) = x^\alpha\sum_{k=0}^{\infty}\frac{(-x)^k}{(\alpha+k)k!}[/tex]

The result is,

[tex]\sum_{k=0}^{\infty}\frac{(-1)^k\beta^{2\alpha}\Gamma(2\alpha+k)}{(\alpha+k)k!}[/tex]

The problem is that it do not always converge, any idea?

Many thanks indeed.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Integartion of Incomplete Gamma Function

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**