I am trying to calculate the Fourier transform of $$f(x)=\frac{x}{(x^2+1)^2}$$ using the property of Fourier transform. So I am trying to use$$\widehat{g_1(x)g_2(x)}=\frac{1}{2\pi}\widehat g_1 *\widehat g_2(\epsilon)$$ Since we already know $$\widehat {\frac{1}{x^2+1}}=\pi e^{-|\epsilon|}$$ .So first I tried to set $$g_1(x)=g_2(x)=\frac{1}{x^2+1}$$ then after finding the $\widehat{g_1(x)g_2(x)}$ by calculating the convolution. Using the property again to calculate $$\widehat f=\widehat{[g_1(x)g_2(x)]x}$$ But I found this doesn't work since the Fourier integral of $x$ doesn't converge.
So I tried to set $$g_1(x)=\frac{x}{x^2+1}$$ $$g_2(x)=\frac{1}{x^2+1}$$ But I don't know how to calculate the Fourier transform of $$g_1(x)=\frac{x}{x^2+1}$$ and I also don't know if it converges. Since I tried to calculate it by using residue theorem. The integral on the complex part $$\int_{complex}\frac{e^{-iz\epsilon}z}{z^2+1}$$I don't know how to deal with it.
Does anyone know about this. Please help. Thanks so much!:)
You are on the wrong track when trying to work with $\frac{x}{(x^2+1)^2}$ as the product of $\frac{1}{x^2+1}$ and $\frac{x}{x^2+1}$. Instead, use the fact that $$ \frac{d}{dx}\left(\frac{1}{x^2+1}\right) = -\frac{2x}{(x^2+1)^2} $$ and invoke the formula for $\widehat {f'}$ in terms of $\widehat{f}$.