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

[bit188]

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)

Homework Equations

Gamma(n+1) = n Gamma (n)

The Attempt at a Solution

Tried an integration by substitution and an integration by parts, and no luck!

Thanks for your help!

 

[HallsofIvy]

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]

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?

 

[bit188]

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]

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

 

[bit188]

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

Thanks anyhow!

 

[Gib Z]

You sure you don't just want it approximately?

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

 

[bit188]

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