Finding a substitution for an exponential integral

[paco_uk]

Homework Statement

Starting from the Gamma function:

<br /> <br /> \Gamma (s) = \int^{\infty}_{0} dx \, x^{s-1} e^{-x} <br /> <br />

Make a change of variable to express it in the form:

<br /> <br /> \Gamma (s) = f(s) \int^{\infty}_{0} dy \, \exp{\frac{-A(y)}{\zeta(s)}}<br /> <br />And identify the functions f(s), A(y), \zeta(s).

Homework Equations

The Attempt at a Solution

I've tried various solutions along the lines of x^{s} and e^y but I can't find anything that works and I don't know any general methods for finding an appropriate substitution. Can anyone suggest where to start?

 

[fzero]

Have you tried using

x^{s-1} = e^{(s-1) \ln x} ?

I haven't quite solved this but a log substitution seems to almost work.

 

[paco_uk]

Thanks for your suggestion. I think that let's me rewrite the integral as:

<br /> <br /> \Gamma (s)&amp;=&amp;\int^{\infty}_{0} dx \, e^{(s-1) \ln{x}-x}<br /> <br />

but I still can't find a way to get it in the form required. For example, the substitution y= \ln{x} gives:

<br /> <br /> \Gamma (s)&amp;=&amp;\int^{\infty}_{-\infty} dy \, e^{sy -e^{y}} <br /> <br />

... still no use. Is there some clever trick I'm missing?