Dirichlet's problem, conformal mapping and complete sets in Hilbert space.

First Part: One of the proofs of Poisson's formula is analysed. This leads readily to a method of solving Dirichlet's problem explicitly in some new cases. The solution of Dirichlet's problem is equivalent to the conformal mapping of some given simply connected region on the interior of a circle. The new method for the solution of Dirichlet's problem is tested by the conformal mapping of an ellipse on a circle. Thus a result previously found by a different method by Szego is confirmed.