Harmonic Functions — Mathematics Knowledge System

Overview

A real-valued function u(x, y) of two real variables is harmonic on a region if it has continuous second partial derivatives there and satisfies Laplace's equation, ∂²u/∂x² + ∂²u/∂y² = 0. Harmonic functions arise naturally throughout physics as steady-state temperature distributions, electrostatic potentials, and velocity potentials of incompressible irrotational fluid flow. The deep link to complex analysis is that the real and imaginary parts of any holomorphic function are automatically harmonic — a direct consequence of the Cauchy–Riemann equations — and conversely, every harmonic function on a simply connected domain is the real part of some holomorphic function, called its harmonic conjugate. This correspondence lets the entire machinery of complex analysis (conformal mapping, the maximum modulus principle, mean value properties) be brought to bear on two-dimensional potential problems.

Intuition

Laplace's equation says a harmonic function has no local 'bumps': at every interior point its value equals the average of its values on any small circle around that point (the mean value property), so it can have no strict local max or min in the interior of its domain — any hot spot or cold spot must sit on the boundary. That every holomorphic function's real part is harmonic follows immediately from the Cauchy–Riemann equations: differentiating ∂u/∂x = ∂v/∂y with respect to x and ∂u/∂y = −∂v/∂x with respect to y and adding gives ∂²u/∂x² + ∂²u/∂y² = 0, because the mixed partials of v cancel (assuming they're continuous, which holomorphy guarantees). So harmonic functions are exactly the 'shadows' cast onto one coordinate by a holomorphic function — and finding the other coordinate (the harmonic conjugate) amounts to reconstructing the whole holomorphic function from one of its two real parts.

Formal Definition

Definition

A function u : Ω → ℝ on an open set Ω ⊆ ℝ² is harmonic if u ∈ C²(Ω) and satisfies Laplace's equation:

\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0

Laplace's equation

f = u + iv \text{ holomorphic} \;\Longrightarrow\; \nabla^2 u = 0,\ \nabla^2 v = 0

Real and imaginary parts of a holomorphic function are harmonic

u(z_0) = \frac{1}{2\pi}\int_0^{2\pi} u(z_0 + re^{i\theta})\,d\theta

Mean value property

Worked Examples

Compute the first and second partials with respect to x.
$\frac{\partial u}{\partial x} = 2x, \qquad \frac{\partial^2 u}{\partial x^2} = 2$
Compute the first and second partials with respect to y.
$\frac{\partial u}{\partial y} = -2y, \qquad \frac{\partial^2 u}{\partial y^2} = -2$
Sum the second partials.
$\nabla^2 u = 2 + (-2) = 0$

Answer: u = x² − y² is harmonic everywhere on ℝ² (it is Re(z²) for z = x+iy, consistent with z² being entire).

Practice Problems

Difficulty 5/10

Show that u(x, y) = e^x cos y is harmonic.

Difficulty 6/10

Find the harmonic conjugate v(x,y) of u(x,y) = e^x cos y, with v(0,0) = 0.

Difficulty 7/10

Is u(x,y) = x³ − 3xy² harmonic? If so, identify a holomorphic function it could be the real part of.

Quiz

Summary

u(x,y) is harmonic if it has continuous second partials and satisfies Laplace's equation ∂²u/∂x² + ∂²u/∂y² = 0.
The real and imaginary parts of any holomorphic function are harmonic, a direct consequence of the Cauchy–Riemann equations.
On a simply connected domain, every harmonic function u has a harmonic conjugate v (unique up to an additive constant) making f = u+iv holomorphic.
Harmonic functions satisfy the mean value property and hence cannot attain a strict interior maximum or minimum.

References

WebsiteWikipedia — Harmonic function