Let $K_\alpha=\{0<\varphi<\alpha\}$ be an angle on the plane. Is there an explicit expression for the Green's function of the Neumann problem for the Laplace equation in the domain $K_{3\pi/2}$?
Say for $K_{\pi/2}$ the Green's function can be constructed by the method of images.
Also for the Dirichlet problem it can be obtained using complex analysis. But I haven't met this approach for the Neumann problem.
The approach via conformal map works fine; the map $\phi(z) = z^{\pi/\alpha}$ has nonzero derivative on the boundary (except for the vertex, where we can't talk about normal derivative anyway), so composing a function with zero normal derivative with $\phi$ or $\phi^{-1}$ yields another function with zero normal derivative. So, moving the problem to the halfplane and back yields $$ G(z, w) = \frac{1}{2\pi}\left(\log \left|z^{\pi/\alpha} - w^{\pi/\alpha}\right| + \log \left|z^{\pi/\alpha} + \overline{w^{\pi/\alpha}}\right|\right) $$ for Neumann Green function in $K_\alpha$.