A Little Pulse…

This past semester I was working as T.A in a Mathematical Physics course. One of the main topics studied was Fourier analysis which I consider a really beautiful subject, but also rather cryptic. This area is fulled with mysterious and magical-looking identities, one of them in particular calls my attention every time I use it and I would like to discuss it here. Also, I think it may be useful to show a more complete (but not formal at all) proof of this fact, because once you “believe” it, the derivation of the Fourier transform identities comes as pretty straight forward. I’m assuming here a certain knowledge or intuition about the Dirac delta function and its sampling property Forgetting about suspense, here is the thing I would like to prove:

\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\omega x}\mathrm{d}\omega

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