- #1
mjordan2nd
- 177
- 1
Homework Statement
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]
Homework 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]
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.