A unified solution of Laplace's equation in an arbitrary region

An exact solution for the Laplace's equation with Dirichlet boundary conditions in an arbitrary region is presented in this work. The solution is in the form of an integral in terms of the potential on the boundary, and depends on the geometry of the region. The method is valid for bounded as well as unbounded regions. Poisson's integral formula and Schwarz's integral rep resentation for the half-plane potential problem are obtained as a verification of the approach. The method also presents a generalized mean-value theorem for harmonic functions at any interior point of a circular disk in terms of weighted values at the boundary of the region.