Harmonic Functions and Applications

  • Nakhlé H. Asmar
  • Loukas GrafakosEmail author
Part of the Undergraduate Texts in Mathematics book series (UTM)


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.

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

Personalised recommendations