This is a slightly expanded version of the original notes with very few changes. The principle has remained the same, namely to present an overview of the classical theory at the level of a graduate course. The part called “Preliminaries” is new and its contents were silently taken for granted during the original course. The main material is the Divergence Theorem and Green’s Formula, a short course on holomorphic functions (, since their real parts are the main examples of harmonic functions in the plane and, also, since one of the
central results is the proof of the Riemann Mapping Theorem through potential theory), some basic facts about semi-continuous functions and very few elementary results about distributions and the Fourier transform. Except for the Divergence Theorem, the Arzela-Ascoli Theorem, the Radon-Riesz Representation Theorem and, of course, the basic facts of measure theory and functional analysis, all of which are used but not proved here, all other material contained in these notes is proved with sufficient detail.