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Homework Help: Proving the Shift Theorem in an Inverse Fourier Transform

  1. Nov 10, 2009 #1
    1. The problem statement, all variables and given/known data

    We are asked to prove that if [tex]F(\omega )[/tex] is the fourier transform of f(x) then prove that the inverse fourier transform of [tex]e^{i\omega \beta}F(\omega)[/tex] is [tex]f(x-\beta )[/tex]

    2. Relevant equations

    [tex]F(\omega)=\frac{1}{2\pi}\int^{\infty}_{-\infty}f(x)e^{i\omega x}dx[/tex]
    [tex]f(x)=\int^{\infty}_{-\infty}F(\omega)e^{-i\omega x}d\omega[/tex]

    3. The attempt at a solution

    We want to find the inverse fourier transform of [tex]\int^{\infty}_{-\infty}F(\omega)e^{i\omega \beta}e^{-i\omega x}[/tex]

    Setting [tex]k=\beta - x[/tex]

    [tex]\int^{\infty}_{-\infty}F(\omega)e^{i\omega \beta -x}d\omega = \int^{\infty}_{-\infty}F(\omega)e^{i\omega k}d\omega=f(k)=f(\beta -x)[/tex]

    I seem to have the signs reversed in my answer. Could anyone help me spot where I missed a sign change please. Thanks.
  2. jcsd
  3. Nov 10, 2009 #2
    For some reason my edits aren't showing up...
  4. Nov 17, 2009 #3
    Could anyone help me out?
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