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Harmonic Function — Definition, Formula & Examples

A harmonic function is a twice-continuously-differentiable function whose Laplacian equals zero at every point in its domain. In other words, the sum of all its unmixed second partial derivatives is always zero.

A function u:Ω⊆Rn→Ru: \Omega \subseteq \mathbb{R}^n \to \mathbb{R} of class C2C^2 is harmonic on Ω\Omega if it satisfies Laplace's equation ∇2u=0\nabla^2 u = 0 throughout Ω\Omega, where ∇2u=∑i=1n∂2u∂xi2\nabla^2 u = \sum_{i=1}^{n} \frac{\partial^2 u}{\partial x_i^2}.

Key Formula

∇2u=∂2u∂x12+∂2u∂x22+⋯+∂2u∂xn2=0\nabla^2 u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2} = 0
Where:
  • uu = A twice-continuously-differentiable scalar function
  • x1,…,xnx_1, \ldots, x_n = Independent variables in the domain
  • ∇2\nabla^2 = The Laplacian operator

How It Works

To check whether a given function is harmonic, compute each of its unmixed second partial derivatives and add them together. If the sum equals zero identically on the domain, the function is harmonic. In two variables, you need uxx+uyy=0u_{xx} + u_{yy} = 0. Harmonic functions have the mean value property: the value at any interior point equals the average over any surrounding sphere (or circle in 2D) contained in the domain. In complex analysis, the real and imaginary parts of any analytic function are each harmonic, connecting this concept directly to the Cauchy-Riemann equations.

Worked Example

Problem: Determine whether u(x,y)=x2−y2u(x, y) = x^2 - y^2 is harmonic.
Step 1: Compute the second partial derivative with respect to xx.
∂2u∂x2=∂∂x(2x)=2\frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x}(2x) = 2
Step 2: Compute the second partial derivative with respect to yy.
∂2u∂y2=∂∂y(−2y)=−2\frac{\partial^2 u}{\partial y^2} = \frac{\partial}{\partial y}(-2y) = -2
Step 3: Add them to evaluate the Laplacian.
∇2u=2+(−2)=0\nabla^2 u = 2 + (-2) = 0
Answer: Since ∇2u=0\nabla^2 u = 0, the function u(x,y)=x2−y2u(x,y) = x^2 - y^2 is harmonic.

Why It Matters

Harmonic functions model steady-state heat distribution, electrostatic potentials, and incompressible fluid flow. In complex analysis courses, recognizing harmonic functions lets you construct analytic functions and solve boundary value problems using techniques like conformal mapping.

Common Mistakes

Mistake: Including the mixed partial derivatives (e.g., uxyu_{xy}) in the Laplacian.
Correction: The Laplacian sums only the unmixed (pure) second partial derivatives: uxx+uyyu_{xx} + u_{yy} in 2D. Mixed partials do not appear.