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How to prove the value of Gamma(1/4)?

  1. Feb 12, 2007 #1
    1. The problem statement, all variables and given/known data
    Self-given problem; I want to prove that Gamma (1/4) is approxiamately equal to 3.625, but can't seem to integrate it properly...

    Gamma(z) = (integral between infinity and 0) (t^z-1)(e^-t) dt

    2. Relevant equations
    Gamma(n+1) = n Gamma (n)

    3. The attempt at a solution
    Tried an integration by substitution and an integration by parts, and no luck!

    Thanks for your help!
    Last edited: Feb 12, 2007
  2. jcsd
  3. Feb 12, 2007 #2


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    So you need to integrate
    [tex]\int_0^\infty t^{-3/4}e^{-t}dt[/tex]

    What do you mean by "approxiamately equal to 3.625". If that is really what you want to get that, then use a numerical integration.
  4. Feb 12, 2007 #3

    matt grime

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    2. That is not the only relevant bit of information, but hey, what the heck (it needs to be an analytic continuation).

    3. What makes you think anything but a numerical approximation will work?
  5. Feb 12, 2007 #4
    No, I want to get the real value :P

    When I do an integration by parts, I get

    (-3/4)(t^6/4) + (3/4)(t^-1/2)(e^-t), which is not the right answer...
  6. Feb 13, 2007 #5


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    [itex] \Gamma\left(\frac{1}{4}\right) [/itex] cannot be expressed in terms of values of common transcendental functions.
  7. Feb 13, 2007 #6
    Really? Well, I suppose I'd better learn more math, then. >_<

    Thanks anyhow!
  8. Feb 13, 2007 #7

    Gib Z

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    You sure you don't just want it approximately?

    Numerical integration would be fine for a few decimal places >.< good enough lol
  9. Feb 13, 2007 #8
    Nah... I'm pretty intrigued by this particular function. I'm interested for nonpractical reasons. Thanks though! ^_^
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