Results 1 to 8 of 8

Math Help - Proving one of the Gamma Function's properties ..

  1. #1
    Junior Member BayernMunich's Avatar
    Joined
    Jul 2010
    Posts
    25

    Proving one of the Gamma Function's properties ..

    Hello,

    How can you prove that:

    \Gamma(x) \cdot \Gamma(1-x) = \dfrac{\pi}{sin(\pi x)} ?!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by BayernMunich View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by BayernMunich View Post
    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?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member BayernMunich's Avatar
    Joined
    Jul 2010
    Posts
    25
    Thanks. I found an acceptable proof.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    \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?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member BayernMunich's Avatar
    Joined
    Jul 2010
    Posts
    25
    OMG @@
    I did not reach this level xD
    Thanks anyway
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by Also sprach Zarathustra View Post
    \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
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    Quote Originally Posted by chiph588@ View Post
    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!!!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving properties of bumb function
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 12th 2011, 12:38 PM
  2. Properties of Gamma(ln(x))?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 2nd 2011, 01:17 AM
  3. Proving matrix inverse properties
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: July 17th 2010, 09:01 PM
  4. Proving properties of matrix
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: March 31st 2010, 04:01 AM
  5. Replies: 0
    Last Post: February 21st 2009, 03:32 AM

Search Tags


/mathhelpforum @mathhelpforum