# Proving the Shift Theorem in an Inverse Fourier Transform

1. Nov 10, 2009

### mjordan2nd

1. The problem statement, all variables and given/known data

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 $$f(x-\beta )$$

2. Relevant equations

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

3. 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.

2. Nov 10, 2009

### mjordan2nd

For some reason my edits aren't showing up...

3. Nov 17, 2009

### mjordan2nd

Could anyone help me out?