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Homework Help: Gamma function to Stirling Approximation

  1. Sep 21, 2008 #1
    1. The problem statement, all variables and given/known data

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

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

    3. The attempt at a solution

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

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

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

    Where, if anywhere, did I go wrong?
     
  2. jcsd
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