Integration: Gamma Functions

[Xittenn]

I've yet to come across this despite the numerous references to Gamma functions in various places which I never gave notice. To what I am referring is integration of factorials which is now to my understanding a specific case of the gamma function. What are some of the finer points to be looked over when considering solutions? Where do these concepts lie in the greater scheme of mathematics?

[ajb]

Am I right in thinking you want to evaluate a path integral for the Euler gamma function?

 

The gamma function is analytic over the complex plain except at an the (countably) infinite number of points [math]z= -k | k \in \mathbb{N}[/math] which are simple poles with residues [math] \frac{(-1)^{k}}{k!}[/math].

 

Then knowing this you may be able to use the residue theorem, depending on what you want to do exactly.

 

I know that there are some nice expressions for infinite contours , you will have to look them up.

[Xittenn]

Am I right in thinking you want to evaluate a path integral for the Euler gamma function?

 

This is something I am going over but my original thoughts where more focused on Gamma distribution and probability! I'm going through a slew of information with this one.

[ajb]

Ok, sorry I had the wrong gamma function!

 

You meant this one.

 

As where I thought you meant this one.

 

I think for the gamma function you are interested in is usually called the gamma distribution to avoid such confusions.

[Xittenn]

But it's still expressed in terms of the [math] \Gamma[/math] Function hence the interest!

Edited August 10, 2009 by buttacup

[ajb]

Isn't the shape function always taken to be real and positive?

 

Thus we don't have to worry about the poles.

[Xittenn]

I'm starting to get a clearer picture of what I'm asking.

 

So I'm looking to approach the problem of integrating factorials for use in probability equations. I guess what I was missing here is that to integrate factorials you must first represent it as some function. This led me to Gamma Functions as they are representations of factorials. So to find the integral of a factorial I must find the integral of a Gamma Function. This can be done with some difficulty yes/no? It is to my understanding that approaching integration of Gamma Functions is usually done numerically! Apparently there are other such ways of approaching integration of factorials of which I am now reviewing. I'm sure a lot of this is covered in a book I haven't started which at the moment is not in my presence...............I hope to be going through this book next week. I find it odd that the online material for this stuff is somewhat unavailable, well, as a referenced topic; probability relies heavily on factorials. Is there some reason for this that I'm overlooking?

Edited August 11, 2009 by buttacup

[ajb]

I think there are some nice expressions for certain definite integrals of the gamma function. However, I don't know any of the details here.

[Xittenn]

I'll look through the big book of integration tables, thanks for noting this AJB!