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
You should consider the integral definition of the Gamma function : Eq.(3) in : Gamma Function -- from Wolfram MathWorld
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
I have played with the integral a bit but I'm not getting what I want, I'm going to keep investigating.
Try this: When r is not restricted to an integer value, the more general form is Gamma(r) = integral from 0 to infinity of x^{(r-1)}e^{-x} dx. With some pretty creative substitutions that bring in the relationship of the Gamma to the standard normal distribution, you will find that Gamma(1/2) = Pi^{1/2 }. If you do integration by parts for a few iterations it is easy to see how on Gamma(r) you will see very readily why Gamma(r) = (r-1)! when r is restricted to an integer value...
The solution for Gamma(3/2) can also be derived in similar fashion, but other fractional value for r get very dicey if not impossible to solve with pencil and paper.
Gamma is messy, but Gamma is beautiful too. So many connections to so many other distributions. Simplest connection is that a Gamma is just the sum of independent exponential distributions....then there is the Poisson/Gamma connection which is fascinating to me....gotta learn to love Gamma!
Rather than make a new topic I am posting here because the thread title is also relevant to my question.
Here are three examples of applying the gamma function:
1.
2.
3.
It seems like there is a pattern wherein integration is not necessary to acquire these results.. what were the steps here? there is a way to do it without integrating correct?