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#### skatenerd

##### Active member

- Oct 3, 2012

- 114

Define for any \(r\geq0\) (real):

\(\Gamma(r)=\int_{0}^{\infty}{x^r}{e^{-x}}\,dx\)

a. Show that \(\Gamma(0)=1\)

This one was relatively easy. Just plugged in 0 to the r in the integral and got the answer 1.

b. Show that for any \(r\geq0\):

\(\Gamma(r+1)=(r+1)\Gamma(r)\)

When I tried solving for this all I seemed to be able to figure out was to plug the integral that is equal to \(\Gamma(r)\) into the right side of the equation, but from there I really just have no idea what to do with the \(\Gamma(r+1)\).

and if we can get there...

c. Conclude that for any \(n\in N\) (real):

\(\Gamma(n)=n!\)

I feel like solving b might give insight to this problem, but right now all I can say is that I have no idea how a third variable "n" just came into this problem.