Proving the Shift Theorem in an Inverse Fourier Transform

[mjordan2nd]

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.

 

[mjordan2nd]

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

 

[mjordan2nd]

Could anyone help me out?