Fourier Transform of $xf(x)$

Question

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 .

Answer 1

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 $$FT[xf(x)] = \int_{-\infty}^{\infty} x f(x) e^{i k x} dx $$ $$ = \int_{-\infty}^{\infty} \frac{1}{i} \frac{\partial}{\partial k} \Big[ f(x) e^{i k x} \Big] dx = -i \frac{\partial}{\partial k} \int_{-\infty}^{\infty} f(x) e^{i k x} dx$$ which means $$FT[xf(x)] = -i \frac{\partial f(k)}{\partial k} $$