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- Apr 9th 2011, 02:12 PMTheCoffeeMachine(Basic) Integral
Show that .

- Apr 9th 2011, 06:18 PMProve It
I expect you would have to use Contour Integration and the Residue Theorem to evaluate this...

- Apr 9th 2011, 08:01 PMKrizalid
For of course.

Just put and invoke Beta-Gamma Function.

I wouldn't actually call it as a "basic integral," I could post other stuff which I consider basic for me, but hard for others. - Apr 9th 2011, 08:43 PMTheCoffeeMachine
I'm sorry, I didn't realise it was that hard! I've seen someone use it to calculate a relatively easy sum, so I assumed its proof should at least be easier than the calculation of the sum. It turns out not. I've tried to prove it, but got nowhere, so I assumed I was missing something obvious somewhere, hence the somewhat reckless/misleading title. I'll see what I can do with the beta function.

- Apr 9th 2011, 08:50 PMTheCoffeeMachine
- Apr 9th 2011, 08:52 PMProve It
- Apr 9th 2011, 08:58 PMTheCoffeeMachine
- Apr 9th 2011, 09:27 PMRandom Variable
I have this written down in my notebook. It can be used to evaluate some other integrals.

Let .

Then and

let

and by the definition of the Beta function

now use the reflection formula for the gamma function

- Apr 9th 2011, 09:45 PMRandom Variable
I'm confused how you would use contour integration when the number of residues in the upper half plane is dependent upon the value of . Perhaps there's a more appropriate contour that I'm unware of.

- Apr 9th 2011, 10:39 PMTheCoffeeMachine
Thanks for the proof, RV. Looks like this integral was anything but basic! O_o

- May 30th 2011, 09:44 PMTheCoffeeMachine
See this.