# Gamma function to Stirling Approximation

1. Sep 21, 2008

### Bill Foster

1. The problem statement, all variables and given/known data

Show that the integrand of $$\Gamma(s+1)=\int_{0}^{\infty} t^se^{-t}dt$$ may be written as $$e^{f(t)}$$ where $$f(t)=s\ln{t}-t$$. Show that $$f(t)$$ is maximum at $$t=t_0$$ and find $$t_0$$.

If the integrand is sharply peaked, expand the integrand about this point (ie Taylor expansion) and evaulate the integral to obtain $$\Gamma{(s+1)}=s^se^{-t}\sqrt{2\pi}[1+\frac{1}{12s}+o(\frac{1}{s^2})]$$

3. The attempt at a solution

I can get the first part. However, when I Taylor Expand $$e^{slnt-t}$$ I get the following:

$$s^se^{-s}[1-\frac{1}{2s}(t-s)^2]$$

I don't see how integrating this is going to get the result I seek.

Where, if anywhere, did I go wrong?