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An integral arising from the inverse Fourier transform

  1. Nov 24, 2012 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    For a physics problem I must take the inverse Fourier transform of 2 functions.
    Namely I must compute the integral ##\frac{1}{\sqrt{2\pi}}\int_{-\infty} ^\infty [A\cos (ckt)+B\sin (ckt)]e^{ikx}dk##.


    2. Relevant equations
    Already given.
    i is the complex number. t is greater or equal to 0. In fact it could also be negative, there should be no problem.

    3. The attempt at a solution
    So I tried to tackle ##\int _{-\infty} ^\infty \cos (ckt)e^{ikx}dk## first but ran out of ideas.
    Integration by parts does not look promising. Probably some substitution I guess but I don't see it. Any tip would be appreciated.
     
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  3. Nov 24, 2012 #2

    vela

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    Look into the integral representation of the Dirac delta function.
     
  4. Nov 24, 2012 #3

    fluidistic

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    Hi and thanks for the tip vela. I have it under the eyes (http://dlmf.nist.gov/1.17), but I don't see how this can help.
    Edit: I rewrote ##A\cos (ckt)+B\sin (ckt)## as ##Ae^{ickt}+Be^{-ickt}## which makes the integral diverge.
     
    Last edited: Nov 24, 2012
  5. Nov 24, 2012 #4

    vela

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    The integrals don't converge in the normal sense, but you can use 1.17.12 to recognize the appearance of the delta function.
     
  6. Nov 24, 2012 #5

    vela

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    If you don't find that approach satisfying, you can try throwing in a convergence factor and then taking a limit:
    $$\lim_{\lambda \to 0^+} \int_{-\infty}^\infty \cos (ckt) e^{ikx} e^{-\lambda |k|}\,dk$$ You have to be a bit careful when taking the limit so that you end up with the delta functions.
     
  7. Nov 24, 2012 #6

    fluidistic

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    Thanks a lot! It does satisfy me. So the answer would be ##A\delta (ct+x)+B\delta (ct-x)## where A and B are not necessarily the constants I started with.
    I hope it's right.
     
  8. Nov 24, 2012 #7

    fluidistic

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    I just found out the solution to the problem. Apparently A and B depend on k (http://mathworld.wolfram.com/WaveEquation1-Dimensional.html), I missed this. The answer should be f(x-ct)+f(x+ct) instead of the delta. f could be any function twice differentiable I think, thus the delta is a possibility.
    P.S.:I made a typo in my previous post. The argument of the second delta should be "x-ct".
     
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