  In the context of the correspondence between real functions on the unit
circle and inner analytic functions within the open unit disk, that was
presented in previous papers, we show that the constructions used to establish
that correspondence lead to very general proofs of existence of solutions of
the Dirichlet problem on the plane. At first, this establishes the existence of
solutions for almost arbitrary integrable real functions on the unit circle,
including functions which are discontinuous and unbounded. The proof of
existence is then generalized to a large class of non-integrable real functions
on the unit circle. Further, the proof of existence is generalized to real
functions on a large class of other boundaries on the plane, by means of
conformal transformations.
