(i) Show that the inverse Fourier transform of the function

$$\hat{g}(s)= \begin{cases}e^{s}-e^{-s}, & |s| \leqslant 1 \\ 0, & |s| \geqslant 1\end{cases}$$

is

$$g(x)=\frac{2 i}{\pi} \frac{1}{1+x^{2}}(x \sinh 1 \cos x-\cosh 1 \sin x)$$

(ii) Determine, by using Fourier transforms, the solution of the Laplace equation

$$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0$$

given in the strip $-\infty<x<\infty, 0<y<1$, together with the boundary conditions

$$u(x, 0)=g(x), \quad u(x, 1) \equiv 0, \quad-\infty<x<\infty$$

where $g$ has been given above.

[You may use without proof properties of Fourier transforms.]