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Integrals for the LogGamma & polygamma fcns

  1. Jun 3, 2006 #1

    benorin

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    I have proved that for [tex]\Re [z]>0,[/tex]

    [tex]\log \Gamma (z) = \int_{0}^{\infty} \left( z-1+\frac{1-e^{-(z-1)t}}{e^{-t}-1}\right) \frac{e^{-t}}{t}dt[/tex]​

    and I wish to justify differentiating under the integral sign (n+1)-times to give integral formulas for the polygamma functions. Let

    [tex]f(t)=\left( z-1+\frac{1-e^{-(z-1)t}}{e^{-t}-1}\right) \frac{e^{-t}}{t}[/tex]​

    I have proved that if [tex]\Re [z]>0,\mbox{ and }|z|<\infty,[/tex] then

    [tex]\lim_{t\rightarrow 0^+} f(t)=\frac{1}{2}z^2-\frac{3}{2}z+1,\mbox{ and }\lim_{t\rightarrow\infty} f(t)=0[/tex]​

    and if I recall correctly, it is sufficient that the integral in question be absolutely convergent to apply the differentiation rule.

    EDIT: So how do I absolute convergence? Any hints?
     
    Last edited: Jun 3, 2006
  2. jcsd
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