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

1. Feb 12, 2007

### bit188

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
(http://www.wikipedia.org/wiki/Gamma_function)

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!

Last edited: Feb 12, 2007
2. Feb 12, 2007

### 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.

3. Feb 12, 2007

### 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?

4. Feb 12, 2007

### 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...

5. Feb 13, 2007

### dextercioby

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

6. Feb 13, 2007

### bit188

Thanks anyhow!

7. Feb 13, 2007

### Gib Z

You sure you don't just want it approximately?

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

8. Feb 13, 2007

### bit188

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