holomorphic functions and Laplace’s equation

It can be easily shown that the real and imaginary partsDlmfPlanetmath of any holomorphic functionsMathworldPlanetmath (that is, functionsMathworldPlanetmath satisfying the Cauchy-Riemann equationsMathworldPlanetmath) separately satisfy Laplace’s equation. Consider the Cauchy-Riemann equations:


Now differentiate the first equation with respect to x and the second with respect to y:


Now add both equations together:


v must be continuousMathworldPlanetmath, as it is holomorphic, and the mixed derivatives of continuous functions are equal. Hence:


The same process, repeated for v, yields Laplace’s equation for v. u and v are harmonic functionsMathworldPlanetmath, as they satisfy Laplace’s equation, and they are referred to as conjugate harmonics. Functions satisfying Laplace’s equation are important in electromagnetism, and the search for harmonic functions forms part of potential theoryMathworldPlanetmath.

Title holomorphic functions and Laplace’s equation
Canonical name HolomorphicFunctionsAndLaplacesEquation
Date of creation 2013-03-22 17:47:32
Last modified on 2013-03-22 17:47:32
Owner invisiblerhino (19637)
Last modified by invisiblerhino (19637)
Numerical id 7
Author invisiblerhino (19637)
Entry type Application
Classification msc 30E99
Related topic PotentialTheory