Simplifying Gamma(xy) Expression - Help Appreciated

[mmzaj]

what is the simplification of the following expression (in terms of gamma and\or other functions) ?

$$\Gamma(xy)$$

i tried the following :

$$\Gamma(xy)=\int^{\infty}_{0} t^{xy-1} e^{-t} dt$$

now let $$t^x = s$$

=> ( after some manipulation )

$$\Gamma(xy)=\frac{1}{x}\int^{\infty}_{0} s^{y-1} exp{-s^\frac{1}{x}} ds$$

but here is where I'm stuck .. so any help would be appreciated ..

 

[D H]

What do you think it is? We don't do your homework here. You have to show us your work and we help you get past difficulties.

 

[mmzaj]

u r absolutely right , my bad ! by the way , it's not a H.W , it's for a research !

 

[benorin]

Gauss multiplication formula

 

[mmzaj]

thanx , i was aware of qauss multiplication theorem , but it constrains either x or y to be an integer , while I'm looking for the case where neither is !

 

[DavidWhitbeck]

How could you simplify $\Gamma (xy)$ further?? What do you mean by simplify if that's not good enough for you? Any identity that you can invoke will just make it more complicated. Is there something specific you had in mind?

 

[mmzaj]

yup , what i meant is separating (xy) , any other form - regardless of it's complexity - that separates (xy) as an argument is good enough .

 

[benorin]

You are not likely to find it, but I looked and I would not have been deterred by such a statement...

A good reference for formulas is http://functions.wolfram.com, in particular the gamma function (Use the 'PDF file' link on the left-hand plane,) and related functions.

The infinite product definitions for the gamma and beta functions are defined everywhere the functions can be defined (most of the complex plane, e.g. for the gamma function in complex plane less the non-negative integers.)