Harmonic Functions and Applications

Abstract

There are many important applications of complex analysis to real-world problems. The ones studied in this chapter are related to the fundamental differential equation

$$\begin{aligned} \varDelta u=\frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}=0, \end{aligned}$$

known as Laplace’s equation. This partial differential equation models phenomena in engineering and physics, such as steady-state temperature distributions, electrostatic potentials, and fluid flow, just to name a few. A real-valued function that satisfies Laplace’s equation is said to be harmonic. There is an intimate relationship between harmonic and analytic functions. This is investigated in Section 6.1 along with other fundamental properties of harmonic functions.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way.

-Godfrey Harold Hardy (1877–1947)