# Integral Representation of Gamma Function

• Apr 4th 2010, 04:15 AM
EinStone
Integral Representation of Gamma Function
Let $\displaystyle \Gamma_N(s) = \frac{N^s(N-1)!}{s(s+1)\cdot \ldots \cdot (s+N-1)}$.

Show, for every $\displaystyle N \in \mathbb{N}$ that

$\displaystyle \int_0^N \displaystyle \left(1 - \frac{t}{N}\right)^N t^{s-1}dt = N^s \sum_{k=0}^N \frac{(-1)^k {N \choose k}}{k+s} = \frac{N}{N+s}\Gamma_N(s)$.

Hint: For the second inequality, divide both sides by $\displaystyle N^s$ and compare the resulting rational funtions in terms of their poles.
• Apr 4th 2010, 10:24 AM
chiph588@
Quote:

Originally Posted by EinStone
Let $\displaystyle \Gamma_N(s) = \frac{N^s(N-1)!}{s(s+1)\cdot \ldots \cdot (s+N-1)}$.

Show, for every $\displaystyle N \in \mathbb{N}$ that

$\displaystyle \int_0^N \displaystyle \left(1 - \frac{t}{N}\right)^N t^{s-1}dt = N^s \sum_{k=0}^N \frac{(-1)^k {N \choose k}}{k+s} = \frac{N}{N+s}\Gamma_N(s)$.

Hint: For the second inequality, divide both sides by $\displaystyle N^s$ and compare the resulting rational funtions in terms of their poles.

Here's a hint for the first equality:

$\displaystyle \left(1-\frac{t}{N}\right)^N = \sum_{k=0}^N {N\choose k} \left(-\frac{t}{N}\right)^k$ by the binomial expansion theorem.
• Apr 5th 2010, 01:27 AM
EinStone
I don't see how this can help me... :( , can you give further help?

EDIT: Ah lol, I just saw that I can pull everything out of the integral then its easy. Ok I'll now try the second equation, if you can give any help please do it :).