I have: $$g_\epsilon(x)=\sin(x)e^{-\epsilon|x|}$$ of which I want to compute the $F$-transform. Now, I know that singularly the two components of the product function have their own transforms: $$\hat{\sin(x)}=i\pi[\delta(\xi-1)-\delta(\xi+1)]$$ $$\hat e^{-\epsilon|x|}={2\epsilon \over \epsilon^2+\xi^2}$$
The question is, do I really have to manually go through the whole $$\int_Rg_\epsilon(x)e^{-i\xi x}dx$$
or can I use the individual transforms in a smarter way to make it easier for myself?
Thanks
Based on the duality property, the Fourier transform of $x(t)y(t)$ is ${1\over 2\pi}X(j\xi)*Y(j\xi)$ where $*$ denotes the convolution. Since $$\sin x\iff {\pi\over j} [\delta(\xi-1)-\delta(\xi+1)]\\e^{-\epsilon |x|}\iff {2\epsilon\over \xi^2+\epsilon^2}$$we obtain $$e^{-\epsilon |x|}\sin x\iff {\epsilon\over j} \left[{1\over (\xi-\epsilon)^2+\epsilon^2}-{1\over (\xi+\epsilon)^2+\epsilon^2}\right]$$