Solve $∇^2(u) = 0$ in two dimensions for $r < 1$ (in plane polar coordinates), with boundary conditions $u(1, θ) = A$ for $0 < θ < π$ , $0$ for $π < θ < 2π$ where $A$ is a given constant.
I have done the method by separation and found the right equation. Now i am struggling with plugging in the boundary conditions. The bit in particular is the fourier series transformation at the end. Can anyone show a detailed process of the last part.
The separated solutions are $$ A_0, (A_n\cos(n\theta)+B_n\sin(n\theta))r^{n},\;\;\; n=1,2,3,\cdots $$ The coefficients $A_n,B_n$ are determined by Fourier series $$ u(1,\theta)= A_0+\sum_{n=1}^{\infty}(A_n\cos(n\theta)+B_n\sin(n\theta)) $$