# 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!

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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.

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...

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