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How to do fourier transformation of power law functions

  1. Feb 13, 2010 #1
    As the title, I want to know details of the following integrations

    \int |x|^a * exp[i*k*x] * dx = k^{-1-a} * Gamma[1+a] * sin[a*pi/2] -------(1)

    by variable changes, k*x -> z, it's easy to get the factor k^{-1-a}, i.e.

    l.h.s -> (\int z^a * exp[i*z] * dz) / k^{1+a} -------------------------------------(2)

    but the remaining integration seems very difficult.
    We know,

    \int z^a * exp[-z] * dz \propto Gamma[1+a] --------------------------------------(3)

    But, how to do integrations in eq(2) whose exponential argument is
    imaginary instead negative?
  2. jcsd
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