$\Delta f = 0$ in $X$ where $X=\{x \in \mathbb{R}^2:r_1 < |x| < r_2\}$.The boundary condition is $f(r_1,\phi)=v(\phi),f(r_2,\phi)=w(\phi)$.Solve a solution $f$.
My try:I used separation of variables and got a very long and complicated form.But I am curious if there is any other better methods like M.V.P. or something else?