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Let $u(x,y)$ be fourier transformed in x to

$\tilde{u}(k,y)=\pi e^{-|k|(1+y)}$

Then by inverting the transform derive the solution for $u(x,y)$.

I know that $\mathcal{F}(\frac{1}{1+x^2})=\pi e^{-|k|}$. Also $\mathcal{F}({\frac{y}{\pi (y^2+x^2)}})=e^{-|k|y}$. Then using convolutions I get

$\frac{1}{\pi}\int_{- \infty}^{\infty}\frac{y}{y^2+(x-\epsilon)^2} \frac{1}{1+\epsilon^2} d\epsilon$. I cannot see where to go from here.

Trajan
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  • What is the question? What $u(x,y)$ was? – Daniel Fischer Mar 07 '14 at 19:41
  • yes. the question derives $\tilde{u}(k,y)$ then asks you invert it in order to find out what $u(x,y)$ is. – Trajan Mar 08 '14 at 15:04
  • Is $y \geqslant 0$ given? (Or at least $y > 1$?) Without that, $\tilde{u}(k,y)$ grows too fast to have a (nice, at least) Fourier transform. – Daniel Fischer Mar 08 '14 at 15:25
  • at the start of the question (before $\tilde{u}$ was derived) we were given that $u(x,0)=\frac{1}{1+x^2}$ and $u(x,y) \rightarrow 0$ as $x^2+y^2 \rightarrow \infty$ – Trajan Mar 08 '14 at 16:38

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