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Fourier transform from k-space to x

  1. Nov 13, 2012 #1
    I have calculated a k-space function to be f(k) = [itex]\frac{1}{2k}[/itex]

    I want to fourier transform this to find f(x), I have found many different fourier transform equations...can I use this one?

    f(x) = [itex]\frac{1}{\sqrt{2π}}[/itex][itex]\int[/itex][itex]\frac{1}{2k}[/itex]e-ikxdk Limits fo integration -Infinity to Infinity

    I'm also having trouble remembering my complex analysis, would I use residue theorem for this integral? I don't quite remember exactly how that goes...

    ∫f(z) = 2πiRes(f,z0) Correct?

    The residue for f(k) is 1/2, so I am not sure excatly what the answer means? Or if I am using residue theorem correctly?

    Final answer: f(x) = [itex]\frac{1}{\sqrt{2π}}[/itex]2πi(1/2) = [itex]\sqrt{\frac{π}{2}}[/itex]i
    This doesn't make sense as an answer.

    Thanks,
    Jason
     
  2. jcsd
  3. Nov 13, 2012 #2

    Mute

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    Homework Helper

    The issue with your calculation is that you seem to have forgotten that the residue theorem applies when the contour you are integrating over completely encircles a pole in the complex plane. What you calculated is the integral over a contour around the point k = 0. The integral you want to calculate is

    $$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty dk~\frac{e^{-ikx}}{k},$$
    so you need a closed contour, at least one piece of which runs along the real axis while avoiding the pole at k = 0. One typically skirts around the pole with a semi-circular arc of small radius ##\epsilon##, and then closes the contour with an arc of radius R, either in the upper half plane or the lower half plane. One takes ##R \rightarrow \infty## and ##\epsilon \rightarrow 0## at the end of the calculation.

    There are a number of important things to note about such a contour: 1) the pole is not enclosed by it, so the full contour integral is zero; the integrals along the segments of the contour thus sum to zero. 2) Which half-plane (negative or positive) you close the large arc in will depend on whether or not x is positive or negative.

    Have you computed a contour integral like this before? Give it a shot and see what you find.
     
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