61
$\begingroup$

It seems as if no one has asked this here before, unless I don't know how to search.

The Gamma function is $$ \Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx. $$ Why is $$ \Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\ ? $$ (I'll post my own answer, but I know there are many ways to show this, so post your own!)

  • 0
    Possible duplicate: http://math.stackexchange.com/questions/9286/proving-int-0-infty-e-x2-dx-frac-sqrt-pi22012-10-20
  • 4
    @Argon Did you look at the other question? It's not even close to the same thing. And, how on earth did 2 other people vote to close without even looking at it?2012-10-20
  • 6
    @Graphth With a simply change of variables ($x^2 = t$), the integrals are virtually identical, except for a constant coefficient.2012-10-20
  • 0
    Maybe not that question, but I am quite sure this has been asked before...2012-10-21
  • 1
    http://www.math.uconn.edu/~kconrad/blurbs/.../gaussianintegral.pdf is a good source of methods to solve this.2013-03-17

11 Answers 11