Harmonic Functions & Conjugates

Laplace's equation, harmonic parts, and building the conjugate.

The idea

A real-valued function $u(x,y)$ with continuous second partial derivatives is harmonic on an open set when it satisfies Laplace's equation $u_{xx} + u_{yy} = 0$ at every point of the set.

Harmonic functions are exactly the real parts of holomorphic ones. One direction is a short computation: if $f = u + iv$ is holomorphic, the Cauchy-Riemann equations give $u_{x} = v_{y}$ and $u_{y} = -v_{x}$; differentiating the first with respect to $x$, the second with respect to $y$, and adding gives $u_{xx} + u_{yy} = v_{yx} - v_{xy} = 0$, since the mixed partials agree. The same argument shows $v$ is harmonic.

The converse needs a hypothesis on the domain. If $u$ is harmonic on a simply connected open set, then some $v$ on that set, unique up to an additive real constant, makes $u + iv$ holomorphic; such a $v$ is a harmonic conjugate of $u$. To construct it, integrate $v_{y} = u_{x}$ with respect to $y$, which determines $v$ up to an unknown function of $x$ alone, then determine that function from $v_{x} = -u_{y}$.

Simple connectivity cannot be dropped. On the plane with the origin removed, $u = \log\sqrt{x^{2} + y^{2}}$ is harmonic, but a conjugate would have to be the polar angle, which gains $2\pi$ on every loop around the origin and so is not a single-valued function there.

Ways to work on it

Not sure where to start? Take the ten-question placement test.