Fourier Transform of $xf(x)$

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I am not able to get the Fourier Transform of $xf(x)$ if $&ltf(x)>$ is the Fourier transform of $f(x)$ .
BTW i tried using convolution theorem but didn't work out .

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1 Answer

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If the Fourier-transform of $f(x)$ is$$FT[f(x)] \equiv f(k) = \int_{-\infty}^{\infty} f(x) e^{- i k x} dx$$then$$FT[xf(x)] = \int_{-\infty}^{\infty} x f(x) e^{- i k x} dx $$ $$ = \int_{-\infty}^{\infty} i \frac{\partial}{\partial k} \Big[ f(x) e^{-i k x} \Big] dx = i \frac{\partial}{\partial k} \int_{-\infty}^{\infty} f(x) e^{-i k x} dx$$which means$$FT[xf(x)] = i \frac{\partial f(k)}{\partial k} $$

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