# Math Help - Gamma function related to fractions

1. ## Gamma function related to fractions

I am working on on understanding $S_n=\frac{2\pi^\frac{n}{2}}{\Gamma(\frac{n}{2})}$ which is the hyper-surface area of a unit n-sphere

this post is specifically about the gamma function. I just need some direction

I know it can be thought of as $\Gamma(n)=(n-1)!$ but if I am doing n=1/2,3/2,.... I have a fraction so that doesn't work. I want to understand how the gamma function works when n is a fraction

I found the Euler reflection formula $\Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin \pi x}$ and I thought maybe this was what I needed but then trying some rationals in it

i.e. let $x=\frac{1}{2}$

$\Gamma(\frac{1}{2}) = \sqrt{\pi}$ whereas $\Gamma(\frac{1}{2})\Gamma(1-\frac{1}{2}) = \pi$ and in this case one is just the square of the other but that wasn't the case for other rationals like $\frac{1}{4}$

so this is where I am now

2. ## Re: Gamma function related to fractions

Originally Posted by Jonroberts74
I am working on on understanding $S_n=\frac{2\pi^\frac{n}{2}}{\Gamma(\frac{n}{2})}$ which is the hyper-surface area of a unit n-sphere

this post is specifically about the gamma function. I just need some direction

I know it can be thought of as $\Gamma(n)=(n-1)!$ but if I am doing n=1/2,3/2,.... I have a fraction so that doesn't work. I want to understand how the gamma function works when n is a fraction

I found the Euler reflection formula $\Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin \pi x}$ and I thought maybe this was what I needed but then trying some rationals in it

i.e. let $x=\frac{1}{2}$

$\Gamma(\frac{1}{2}) = \sqrt{\pi}$ whereas $\Gamma(\frac{1}{2})\Gamma(1-\frac{1}{2}) = \pi$ and in this case one is just the square of the other but that wasn't the case for other rationals like $\frac{1}{4}$

so this is where I am now
The Gamma function is not a nice function. There might be other "nice" solutions but, for example, Gamma(1/4) is transcendental. (According to W|A anyway.)

-Dan

3. ## Re: Gamma function related to fractions

You should consider the integral definition of the Gamma function : Eq.(3) in : Gamma Function -- from Wolfram MathWorld

4. ## Re: Gamma function related to fractions

Originally Posted by topsquark
The Gamma function is not a nice function. There might be other "nice" solutions but, for example, Gamma(1/4) is transcendental. (According to W|A anyway.)

-Dan
yeah both Gamma(1/4) and [Gamma(1-1/4)Gamma(1/4)] are transcendental. whats also interesting is $\Gamma(1-x)\Gamma(x)\sin \pi x = \pi$ for all non-integer numbers

it's an interesting function

Originally Posted by JJacquelin
You should consider the integral definition of the Gamma function : Eq.(3) in : Gamma Function -- from Wolfram MathWorld
I have played with the integral a bit but I'm not getting what I want, I'm going to keep investigating.