Fourier transform of a function

  • #1
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Main Question or Discussion Point

how could i calculate the fourier transform

[tex] \int_{-\infty}^{\infty}dx \frac{e^{iux}}{(a^{2}+x^{2})^{s}} [/tex]

if i try contour integral i find 2 poles at x=a and x=_a but of order 's' wich can not be an integer, is there another definition or faster way to calculate the Fourier transform of

[tex] (a^{2}+x^{2})^{-s} [/tex] for every real a and s ??
 

Answers and Replies

  • #2
1,796
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I assume you mean not just typing FourierTransform into Mathematica (or Wolfram Alpha) and getting the answer so if s is non-integer, then the integrand has two branch points at [itex]\pm ia[/itex]. If you choose the branch-cuts to be [itex](\pm ia,\pm i\infty)[/itex], then you can integrate over an analytic branch in the upper half-plane with a slit along the branch cut at the imaginary axis with an indentation around the branch point at ia. I think the integral will go to zero over the two semi-circular legs but I'm not sure at present. Then the Fourier Transform component will equal to the negative of (two components along the branch cut plus the indentation around the branch point).
 

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