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How do you derive this alternate form of the gamma function?

  1. Nov 21, 2012 #1
    [itex]\Gamma[/itex](n) = int(0 to infinity)[(x^(n-1))*e^-x]dx

    Show that it can also be written as:

    [itex]\Gamma[/itex](n) = 2int(0 to infinity)[(x^(2n-1))*e^(-x^2)]dx


    I have no idea how to go about this. I have tried integration by parts of each to see if anything relates, but how can you get from an exp(-x) to and exp(-x^2) term?

    Any help would be appreciated.

    (PS apologies, I am unfamiliar with writing formulae using latex)
     
  2. jcsd
  3. Nov 21, 2012 #2

    Mute

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    Homework Helper

    A simple change of variables will do the trick. Take another look at the integrals and see if you can figure out what substitution you should make.

    (Maybe it will help you see it if you use the variable 't' instead of 'x' in the second integral)
     
  4. Nov 21, 2012 #3
    Right, I was getting confused because I thought the x in each was the same.

    But that makes sense, thank you. :)
     
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