Harmonic Functions

Key idea

Solutions of Laplace’s equation.

In depth

A function with Laplacian u_xx + u_yy = 0 is harmonic. Real and imaginary parts of analytic functions are harmonic, and satisfy the mean value property. They model steady-state heat and potentials.

Worked example

u = x² − y² is harmonic: 2 − 2 = 0, and is the real part of z².

Formula

Real-world uses

• Modelling steady heat distribution, electric potential and gravitational fields

Common mistake

Assuming all smooth functions are harmonic.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.