Complex Methods | Part IB, 2004

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

g^(s)={eses,s10,s1\hat{g}(s)= \begin{cases}e^{s}-e^{-s}, & |s| \leqslant 1 \\ 0, & |s| \geqslant 1\end{cases}


g(x)=2iπ11+x2(xsinh1cosxcosh1sinx)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

2ux2+2uy2=0\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0

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

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

where gg has been given above.

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

Typos? Please submit corrections to this page on GitHub.