I need to calculate the inverse Fourier transform of the following functions:

  1. $\displaystyle f(w) = e^{(-\pmb{i}5w)} * {\rm sinc}(2w) $

  2. $\displaystyle g(w) = \frac{\pmb{i}w}{(3+\pmb{i}w)(1+\pmb{i}w)} $

I'm really stuck and I don't know where to start, any help would be appreciated!

  • $\begingroup$ Is this homework? Are you having trouble evaluating the integrals, or setting them up? $\endgroup$ – Brian Mar 4 '11 at 0:44
  • $\begingroup$ What did you try? $\endgroup$ – AD. Mar 4 '11 at 5:06
  • $\begingroup$ The second question can be solved by staring at it and then using the residue theorem. Do you know that? For the first question: maybe you know which function gives the sinc function when Fourier transformed? $\endgroup$ – Fabian Mar 4 '11 at 22:28

Here are some hints:

1) If $e_t(x)= e^{itx}$,what is $\widehat{e_tf}$?

2) What is $\widehat{f'}$?

Also note that it is important to state the exact definition of the Fourier transform if you wish to compute explicit functions.


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