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

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

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HallsofIvy
Science Advisor
Homework Helper
So you need to integrate
$$\int_0^\infty t^{-3/4}e^{-t}dt$$

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
Science Advisor
Homework Helper
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?

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
Homework Helper
$\Gamma\left(\frac{1}{4}\right)$ cannot be expressed in terms of values of common transcendental functions.

Really? Well, I suppose I'd better learn more math, then. >_<

Thanks anyhow!

Gib Z
Homework Helper
You sure you don't just want it approximately?

Numerical integration would be fine for a few decimal places >.< good enough lol

Nah... I'm pretty intrigued by this particular function. I'm interested for nonpractical reasons. Thanks though! ^_^