Fourier inverse/convolution problem

Question

I'm struggling to do part (b) of this problem. I do not know how to start: I'm trying to use the inversion formula, but I don't know what to do with the $e^{|s|}$ part (the other one is the laplace transform of $ e^{-y|x|}$ .

Answer 1

Set $f(x,y) = \pi^{-1} y / (x^2 + y^2)$ and $g(x) = e^{-|x|}$. Then $$ (f \ast g)(x) = \int_{-\infty}^\infty \frac{y}{\pi}\frac{e^{-|t|}}{(x-t)^2 + y^2} dt = u(x,y). $$ Therefore, by the convolution theorem we have $\hat{u}(s,y) = \hat{f}(s,y) \hat{g}(s)$, so we just need to compute $\hat{f}(s,y)$ and $\hat{g}(s)$. Assuming the result of part (a) and the inversion theorem, we find that $\hat{f}(s,y) = e^{-y|s|}$ and $\hat{g}(s) = 2 /(1+s^2)$. Therefore $$ \hat{u}(s,y) = \frac{2 e^{-y|s|}}{1+s^2}.$$ The rest of the problem follows easily.