Proving one of the Gamma Function's properties ..

**BayernMunich** · July 15th 2010, 03:02 PM

Hello,

How can you prove that:

$\Gamma(x) \cdot \Gamma(1-x) = \dfrac{\pi}{sin(\pi x)}$ ?!

**mr fantastic** · July 15th 2010, 03:34 PM

Originally Posted by BayernMunich

Hello,

How can you prove that:

$\Gamma(x) \cdot \Gamma(1-x) = \dfrac{\pi}{sin(\pi x)}$ ?!

This is a well known formula: Reflection formula - Wikipedia, the free encyclopedia

I don't plan to re-invent the wheel. You will find it proved in many textbooks that deal with this sort of stuff (go to your your institute's library) and I have no doubt a Google search will turn up many websites that prove it.

**chiph588@** · July 15th 2010, 05:08 PM

Originally Posted by BayernMunich

Hello,

How can you prove that:

$\Gamma(x) \cdot \Gamma(1-x) = \dfrac{\pi}{sin(\pi x)}$ ?!

Most ways I know use some higher level theorems about the Gamma function. I do however know of a fairly straightforward proof that involves Fourier series.

Are you familiar with Fourier series?

**BayernMunich** · July 16th 2010, 02:55 AM

Thanks. I found an acceptable proof.

**Also sprach Zarathustra** · July 16th 2010, 03:15 AM

$\displaystyle \frac{1}{\Gamma(x)}=xe^{\gamma x}\prod_{n=1}^{\infty} \left(\left(1+\frac{x}{n}\right)e^{-x/n}\right)$

Why is that?

**BayernMunich** · August 24th 2010, 10:07 AM

OMG @@
I did not reach this level xD
Thanks anyway

**chiph588@** · December 14th 2010, 04:42 PM

Originally Posted by Also sprach Zarathustra

$\displaystyle \frac{1}{\Gamma(x)}=xe^{\gamma x}\prod_{n=1}^{\infty} \left(\left(1+\frac{x}{n}\right)e^{-x/n}\right)$

Why is that?

Sorry for bringing this thread back from the dead. I just noticed this question went unanswered. I wrote you the answer (and more!) in this pdf file.

Proving one of the Gamma Function's properties ..-gamma.pdf

**Also sprach Zarathustra** · December 15th 2010, 06:41 AM

Originally Posted by chiph588@

Sorry for bringing this thread back from the dead. I just noticed this question went unanswered. I wrote you the answer (and more!) in this pdf file.

Click image for larger version.

Name: Gamma.pdf
Views: 14
Size: 61.3 KB
ID: 20110

Thank you very much!!!

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