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

  • Thread starter bit188
  • Start date
45
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1. Homework Statement
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
(http://www.wikipedia.org/wiki/Gamma_function)

2. Homework 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:

Answers and Replies

HallsofIvy
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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.
 
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?
 
45
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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...
 
dextercioby
Science Advisor
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[itex] \Gamma\left(\frac{1}{4}\right) [/itex] cannot be expressed in terms of values of common transcendental functions.
 
45
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Really? Well, I suppose I'd better learn more math, then. >_<

Thanks anyhow!
 
Gib Z
Homework Helper
3,344
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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
 
45
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Nah... I'm pretty intrigued by this particular function. I'm interested for nonpractical reasons. Thanks though! ^_^
 

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