Harmonic Functions

Abstract

A complex-valued function h on an open subset Ω of the complex plane C is called harmonic on Ω if h ∈ C²(Ω) and

$$\Delta h \equiv 0$$

((1 - 1))

on Q. Here

$$\Delta h =\frac{\partial^2 x}{\partial x^2}+\frac{\partial^2 h}{\partial y^2}$$

is the Laplacian of h. We often assume that Ω is a region (that is, an open and connected set) even when connectivity is not needed, and we are mainly interested in the case in which Ω is a disk or half-plane.