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Homework Help: Fourier transform truoble

  1. Apr 26, 2012 #1
    I'm trying to find [itex]\frac{1}{2\pi}\int \limits_{-\infty}^{\infty}e^{-itx}\frac{1}{a^2+x^2}\mathrm{d}x[/itex] where 'a' is a constant.

    First I noticed that there is [itex]\frac {\partial \arctan x}{\partial x}[/itex] in this and using a substitute got [itex]\int \limits_0^{\pi / 2}\cos( t \tan x )\mathrm{d}x[/itex] with some constants in the gaps.
    I then remember that I'm working in complex numbers, factored [itex]a^2+x^2[/itex] and got something essentially along the lines of [itex]\int \frac{e^x}{x}\mathrm{d}x[/itex], or maybe rather [itex]\int \limits_0^{\infty} \frac {\cos tx} {a - ix}\mathrm{d}x[/itex].

    I can't integrate either.
     
    Last edited: Apr 26, 2012
  2. jcsd
  3. Apr 26, 2012 #2

    HallsofIvy

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    Science Advisor

    Your integral does not contain "dx" or "dt". Without that we cannot tell what integration you intend. Is the problem
    [tex]\int \frac{e^{-itx}}{a^2+ x^2} dx[/tex]
    or is it
    [tex]\int \frac{e^{-itx}}{a^2+ x^2}dt[/tex]
    ?
     
  4. Apr 26, 2012 #3
    Oh, sorry. It's dx. I'll fix it right away.
     
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