Find the inverse Fourier transform $G(x)$ of the function

$$g(k)=e^{-a|k|}, \quad a>0, \quad-\infty<k<\infty .$$

Assuming that appropriate Fourier transforms exist, determine the solution $\psi(x, y)$ of

$$\nabla^{2} \psi=0, \quad-\infty<x<\infty, \quad 0<y<1$$

with the following boundary conditions

$$\psi(x, 0)=\delta(x), \quad \psi(x, 1)=\frac{1}{\pi} \frac{1}{x^{2}+1}$$

Here $\delta(x)$ is the Dirac delta-function.