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Integral from 0 to ∞ with singularity at x=0

  1. Feb 2, 2013 #1
    Here's an integral that is currently giving me grey hairs:

    [itex]\int_0^{\infty} \frac{1}{x} \exp(i \frac{k}{x}(a-c \cos(\theta + wx))) dx[/itex]

    I've tried different approaches like contour integration around [itex] x=0 [/itex] and replacing the exponential by its Taylor sum to have:

    [itex]\int_0^{\infty} \sum_{n=0}^{\infty} \frac{1}{x^{n+1}\;n!} (i k (a-c \cos(\theta + wx)))^n dx[/itex]

    I can do the integrals of the even terms by [itex]\int_0^{\infty} = \frac{1}{2}\int_{-\infty}^{\infty}[/itex] and residues, but I don't know how to handle the odd terms.

    Going to sum-extremes I've rewritten the integral to a form where I only need to do integrals of the type

    [itex]\int_0^{\infty} \frac{1}{x^{n+1}} e^{imwx} dx [/itex]

    with m a positive or negative integer, but even here I'm stuck. Please help!
     
  2. jcsd
  3. Feb 2, 2013 #2

    rollingstein

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    Gold Member

    Maybe this is a stupid question, but is the integration over the real axis?
     
  4. Feb 2, 2013 #3
    Yes, x is real. But since the integrand goes to zero for [itex]x\rightarrow \infty [/itex] the direction of integration in the complex plane shouldn't alter the integral...
     
  5. Feb 2, 2013 #4
    Hmm.
    Is it possible that the latter of the three integrals is simply the gamma-function?
     
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