Harmonic functions from a complex analysis viewpoint. (English) Zbl 0604.31001

This very readable article, aimed at the non-specialist, shows how several properties of harmonic functions in plane domains can be obtained using only basic concepts and theorems from complex analysis. The author’s approach is based on the following (known) result, for which an elementary proof is given.
”Logarithmic Conjugation Theorem”: Suppose \(\Omega\) is a finitely connected region, with \(K_ 1,...,K_ N\) denoting the bounded components of the complement of \(\Omega\), and let \(a_ j\in K_ j\) for each j. If u is a (real-valued) harmonic function on \(\Omega\), then there exists an analytic function f on \(\Omega\) and real numbers \(c_ 1,...,c_ N\) such that \[ u(z)=Re f(z)+c_ 1 \log | z-a_ 1| +...+c_ N \log | z-a_ N| \quad (z\in \Omega). \] This is then used to show that an isolated singularity of a bounded harmonic function is removable, to solve the Dirichlet problem for annuli, and to provide a proof of the conformal mapping theorem for doubly connected regions.
Reviewer: S.Gardiner


31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30C20 Conformal mappings of special domains
Full Text: DOI