The Dirichlet problem for the Laplace equation in a starlike domain of a Riemann surface

We consider the Dirichlet problem for the Laplace equation in a starlike domain, i.e. a domain which is normal with respect to a suitable polar co-ordinates system. Such a domain can be interpreted as a non-isotropically stretched unit circle. We write down the explicit solution in terms of a Fourier series whose coefficients are determined by solving an infinite system of linear equations depending on the boundary data. Numerical experiments show that the same method works even if the considered starlike domain belongs to a two-fold Riemann surface.