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A question related with propagators in position space

  1. Feb 14, 2010 #1
    Dear Colleagues,

    How to prove Fourier transformation rules of the following function

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

    I need this equation to prove some conclusions in translating propagators
    in momentum space to position-space. I wish those know solutions to this
    question give me hints or references urgently.

    The following is the information I know, but I cannot get to the final goal.
    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) / x^{1+a} -------------------------------------(2)

    but the remaining integration seems very difficult.
    We know,

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

    So, my key question is, how to do integrations in eq(2) whose exponential
    argument is imaginary instead negative?

    Thanks to everyone!
  2. jcsd
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