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I have an exam next week and I was hoping someone might be able to help me out with this question. Show that the Fourier transform of the function $f(t+a)$ is $e^{iwa}\hat{f}(w)$ . There is a list of 10 of these types of proofs in the book PDE by Walter Strauss. I just can't seem to get started with it.

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  • $\begingroup$ Then, what is your problem? It is not clear what are you asking! Please, improve your question. $\endgroup$ – the_candyman May 7 '16 at 17:53
  • $\begingroup$ Hi Candyman. Sorry I just want to try and prove it. On page 346 of Strauss book there is a table we a few different transformation and this is one that I am stuck with. $\endgroup$ – user147825 May 7 '16 at 18:02
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The Fourier transform of $g(t) = f(t+a)$ is:

$$\hat{g}(w) = \int_{-\infty}^{+\infty} f(t+a) e^{-iwt}dt$$

Perform a change of variable $s = t+a$. Then:

$$\hat{g}(w)= \int_{-\infty}^{+\infty} f(s) e^{-iw(s-a)}ds = e^{iwa}\int_{-\infty}^{+\infty} f(s) e^{-iws}ds = e^{iwa}\hat{f}(w).$$

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