Proof of Shift Theorem: e^{iwv} F(w) = f(x - v)

[topsquark]

Are you telling me to violate the definition and force it to include [imath]e^{i\omega \nu} F(\omega)[/imath]?

How is taking the Fourier transform of [imath]e^{i \omega \nu } F( \nu )[/imath] where [imath]F( \nu )[/imath] is the inverse Fourier transform of [imath]f( \omega )[/imath] violating anything? You simply put [imath]e^{i \omega \nu } F( \nu )[/imath] into the formula for the Fourier transform, just like you did on the line below. It's a substitution, not anything against the rules.

Or this?
[imath]\int_{-\infty}^{\infty} \left ( e^{i\omega \nu} F(\omega) \right ) e^{-i\omega x} \ d\omega[/imath]

(Parentheses added for emphasis.)

I am not telling you to give me the answer for no effort. I have already tried my best but failed. By giving me the answer, I will be able to solve any problem with shifting. If you don't want to help me, just say it. It seems that you are right, I should have not bothered you and searched in somewhere else.

If you want to learn something then try to do the integral above. It's not that hard if you apply the definition of a Fourier transform. It's a one-liner. (Hint: Combine the exponentials. What does this look like?)

Don't give me the fish. Teach me how to fish.

But you keep demanding that we just give you the fish...

-Dan

 

[Steven G]

Thank you very much Steven G for telling me this story. It really means something. I don't know if you added some Drama or it is fully real, but it has affected on me.

Your story reminds me by an old saying.
Don't give me the fish. Teach me how to fish.

The story is 100% true. I will never forget this story as it affected me as well.