- #1
mjordan2nd
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Homework Statement
We are asked to prove that if F(\omega ) is the Fourier transform of f(x) then prove that the inverse Fourier transform of e^{i\omega \beta}F(\omega) is f(x-\beta )
Homework Equations
F(\omega)=\frac{1}{2\pi}\int^{\infty}_{-\infty}f(x)e^{i\omega x}dx
f(x)=\int^{\infty}_{-\infty}F(\omega)e^{-i\omega x}d\omega
The Attempt at a Solution
We want to find the inverse Fourier transform of \int^{\infty}_{-\infty}F(\omega)e^{i\omega \beta}e^{-i\omega x}
Setting k=\beta - x
\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)
I seem to have the signs reversed in my answer. Could anyone help me spot where I missed a sign change please. Thanks.