Introduction

In previous papers [#!CAoRFI!#,#!CAoRFII!#,#!CAoRFIII!#,#!CAoRFIV!#] we have shown that there is a correspondence between, on the one hand, integrable real functions, singular Schwartz distributions and non-integrable real functions which are locally integrable almost everywhere, and on the other hand, inner analytic functions within the open unit disk of the complex plane. This correspondence is based on the complex-analytic structure within the unit disk of the complex plane, which we introduced in [#!CAoRFI!#]. In order to establish this correspondence for integrable and non-integrable real functions, we presented in [#!CAoRFI!#] and [#!CAoRFIV!#] certain constructions which, given just such a real function, produce from it a unique corresponding inner analytic function.

In this paper we will show that these constructions have, as collateral consequences, the establishment of very general constructive proofs of the existence of the solution of the Dirichlet boundary value problem for the Laplace equation on regions of the plane. We will first establish the proof for integrable real functions on the unit circle, then generalize it to non-integrable real functions which are locally integrable almost everywhere on that circle. Furthermore, with the use of conformal transformations it is possible to generalize the proof to integrable and non-integrable real function on other boundaries on the plane. We will first establish this generalization for a large class of differentiable curves, and then, with a single weak additional limitation on the real functions, for a large class of curves that can be non-differentiable at a finite set of points, such as polygons.

For ease of reference, we include here a one-page synopsis of the complex-analytic structure introduced in [#!CAoRFI!#]. It consists of certain elements within complex analysis [#!CVchurchill!#], as well as of their main properties.