Solving an integral equation by Fourier transform

Question

I have to solve the following exercise. Using the Fourier transform to solve the integral equation $$ \frac{1}{1+x^2}=\int_{-\infty}^{\infty}f(y)\frac{1}{x-y}dy. $$

Some hints?

Answer 1

Take $h(x)={1\over x}$ therefore $$f(x)*h(x)=\frac{1}{1+x^2}$$where $*$ denotes convolution operator. Taking Fourier transform from both sides we get:$$F(\omega)H(\omega)=\pi e^{-|\omega|}\to F(\omega)=isgn(\omega)e^{-|\omega|}=ie^{-\omega}u(\omega)-ie^{\omega}u(-\omega)\to \\f(x)=-\frac{1}{\pi}\frac{x}{1+x^2}$$