AbstractExact solution in closed form is obtained to the following mixed problem for a charged sphere: an arbitrary charge density distribution is prescribed at the surface of a spherical cap while an arbitrary potential is given at the rest of the sphere. A new method is introduced making the procedure straightforward and elementary, no special functions or integral transforms involved. Some further applications of the method are discussed. A new type of solution is obtained for the Dirichlet problem with discontinuous boundary conditions. Several examples are considered.