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Where did this term come from? (Fourier)

  1. Aug 17, 2008 #1
    A simple question.

    Suppose I have [tex]\epsilon^2 y''' - y' = \frac{1}{1+x^2}[/tex].

    The goal is to calculate the fourier transform of [tex]y(x,t)[/tex] where we define,

    [tex]\hat{y}(k,t) = F[\phi] = \int_{-\infty}^{\infty} y e^{ikx} ds[/tex]

    We're also given that,

    [tex]F\left[ \frac{1}{1+x^2} \right] = \pi e^{-|k|}[/tex]

    Now we take transforms of both sides:

    [tex]\rightarrow F[\epsilon^2 y''' - y] = F[\frac{1}{1+x^2}][/tex]

    [tex]\rightarrow -i k^3 \epsilon^2 \hat{y} - ik \hat{y} = \pi e^{|k|}[/tex]

    [tex]\rightarrow \hat{y} = -\frac{\pi e^{-|k|}}{ik(1-\epsilon^2 k^2)}[/tex]

    The answer, however, is supposed to be:

    [tex]\hat{y} = -\frac{\pi e^{-|k|}}{ik(1-\epsilon^2 k)} + 2\pi a \frac{k\delta(k)}{k(1-\epsilon^2 k^2)}[/tex]

    where 'a' is some constant.

    My question is why? I know it has something to do with an additive constant, but I need someone to be explicit with the mistake.
  2. jcsd
  3. Aug 17, 2008 #2
    I've figured out a way to 'see' the result:

    Instead write the equation like this:

    [tex]\epsilon^2 y''' - y' = \frac{1}{1+x^2} + \frac{d}{dx} 2\pi{a}[/tex]

    and the result follows automatically.

    But what I don't understand is why I 'need' to do this. Why haven't I seen this in any texts on Fourier transforms (that if one is applying the transform to a (for example), third order equation reducible to second order, one needs to add arbitrary constants.
    Last edited: Aug 17, 2008
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