Finding a substitution for an exponential integral

[paco_uk]

Homework Statement

Starting from the Gamma function:

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

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

$$<br /> \Gamma (s) = f(s) \int^{\infty}_{0} dy \, \exp{\frac{-A(y)}{\zeta(s)}}<br />$$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:

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

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

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

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