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Integral (related to Laplace transform)

  1. May 1, 2012 #1

    Pzi

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    Hi.

    1. The problem statement, all variables and given/known data
    [tex]\int\limits_0^{ + \infty } {\frac{{dt}}{{{e^{st}} \cdot (1 + {e^t})}}} [/tex]

    2. Relevant equations
    The same as...
    [tex]Laplace\left[ {\frac{1}{{1 + {e^t}}}} \right][/tex]

    3. The attempt at a solution
    Found no elegant properties related to Laplace transform here.
    So figured my best shot would be to integrate directly...

    Any suggestions?
     
  2. jcsd
  3. May 1, 2012 #2

    Ray Vickson

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    Homework Helper

    I think it is a non-elementary integral. By changing variables to u = exp(t), and recognizing that for s > 0 we have exp(-s*t) = exp(-s*ln(u)) = u^(-s), the integral becomes
    [tex]J = \int_1^{\infty} \frac{ u^{-s-1}}{1+u} \, du,[/tex] which Maple 11 evaluates as
    [tex] J = -\text{LerchPhi}(-1,1,-s) - \pi \csc(\pi s),[/tex]
    where LerchPhi is the function defined as
    [tex] \text{LerchPhi}(z,a,v) = \sum_{n=0}^{\infty} \frac{z^n}{(v+n)^a} [/tex] if [itex] |z| < 1[/itex] or [itex] |z|=1[/itex] and [itex] \text{Re}(a) > 1.[/itex] It is extended to the whole complex z-plane by analytic continuation.

    RGV
     
  4. May 2, 2012 #3

    Pzi

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    Yea, looks like it's indeed a non-elementary case here...
    As this is a part of some differential equations I will work around it (using convolution), but it will be ugly.

    Thanks for clarification!
     
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