Harmonic Functions
Marvin Rosenblum, James Rovnyak · Birkhäuser Basel eBooks · 1994
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 (1 - 1) $$\Delta h \equiv 0$$ 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.