The function I have is f(x) = x * $\exp(-x^2)$ but the problem is I don't have any domain of integration for it so i assume you take (-$\infty$,$\infty$) ... As follows I test the party of the function and find out is and odd function thus makeing the F($\omega$) = 2 * $ \int_0^\infty$ f(t) * cos($\omega$*t) dt ... Now comes my dilema , how do I solve it ? I've tried it using the propriety of Fourier transformation takeing speratly the Fourier transformation of x and of $\exp(-x^2)$ , got nothing good ...
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$f$ is very close to be the derivative of a Gaussian function. Fourier transform has nice properties with respect to derivation and Gaussian functions are especially nice as well. Did you look in this direction? – KeiOh May 09 '20 at 14:28
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Nope I didn't learn such thing , can you explain please ? – VlAd TbK May 09 '20 at 14:29
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Honestly the wikipedia page is probably going to give you a better feedback: https://en.wikipedia.org/wiki/Fourier_transform#Derivation (but actually the whole page is pretty good). If you actually need to show this special case of the known properties, I think I would first compute the Fourier transform of $g(x) = \exp(-x^2)$ to see for yourself, and then go by integration by part here to get the transform of the derivative. – KeiOh May 09 '20 at 14:43
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In general, there may be some assumptions (tricky if you're starting in this area) to make on the regularity of $f$, although here the Gaussian has everything you would like: decay is exponentially fast at infinities, smooth as much as you want. See also https://math.stackexchange.com/questions/430858/fourier-transform-of-derivative/ – KeiOh May 09 '20 at 14:49