# Fourier Transform of $xf(x)$

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

• Vedananda, could you perhaps expand on that a bit? It's not completely clear what you question is, at least not to me. – Terry Bollinger Apr 22 '12 at 13:10
• Are you asking how to find FT of xf(x) given FT of f(x) is known? Its been while I read Fourier transform but I think the derivation can be found in any book. – user29679 Apr 22 '12 at 13:18
• Yes exactly Anuragsn , if i know what FT of $f(x)$ is , how do i find FT of $xf(x)$. :) – Theorem Apr 22 '12 at 13:25
• Oh... I get it now. But it is already answered... And you should post such stuff to mathematical forums... – Pygmalion Apr 22 '12 at 13:42

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 $$FT[xf(x)] = i \frac{\partial f(k)}{\partial k}$$
• I know it is possible to proceed like the previous answer if $f \in \mathbb{L}^1$ or $f \in \mathcal{S}$ (Schwartz space). Is it possible to apply the same result to other kind of functions, for example $f(x)=\frac{x}{1+x^2}$? – Cleto Pereira May 19 '20 at 19:22