---
title: Harmonic Functions
description: Harmonic functions are twice-differentiable real-valued functions that satisfy Laplace's equation. Learners will understand their connection to analytic complex functions, explore the mean value property, and solve boundary value problems in physics.
category: mathematics
subcategory: complex-analysis
difficulty: beginner, intermediate, advanced
url: /subject/harmonic-functions
---

# Harmonic Functions

Harmonic functions are twice-differentiable real-valued functions that satisfy Laplace's equation. Learners will understand their connection to analytic complex functions, explore the mean value property, and solve boundary value problems in physics.

## Available Resources

1 Books • 2 Courses • 4 Websites

## Websites

### 1. nLab: Harmonic Function

Comprehensive math encyclopedia entry recommended in r/math

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/harmonic+function

### 2. Harmonic Functions - Wikipedia

Wikipedia reference article on harmonic functions, solutions of Laplace's equation, covering the mean value property, maximum principle, Harnack's inequality, Liouville's theorem, and connections to holomorphic functions and potential theory. Useful for quickly checking definitions, standard examples, and core theorems.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://en.wikipedia.org/wiki/Harmonic_function

**Tags:** harmonic-functions, laplace-equation, mean-value-property, potential-theory, complex-analysis

### 3. Terence Tao's Analysis Notes

**Author:** Terence Tao

Graduate lecture notes from Terence Tao's UCLA course on harmonic and Fourier analysis, covering maximal inequalities, Calderon-Zygmund and Littlewood-Paley theory, Sobolev spaces and pseudodifferential operators, with homework and a take-home final for practice.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.math.ucla.edu/~tao/247a.1.06f/

**Tags:** harmonic-analysis, fourier-analysis, calderon-zygmund-theory, littlewood-paley-theory, sobolev-spaces

### 4. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Complex Variables with Applications (MIT 18.04)

**Author:** Jeremy Orloff

Complex analytic functions and their applications: Cauchy's theorem and integral formula, Taylor and Laurent series, residues for evaluating hard integrals, harmonic functions, conformal maps, two-dimensional fluid flow, and Laplace and Fourier transforms. Provides 14 lecture note files plus problem sets and exams with solutions.

**Difficulty:** Intermediate | **Price:** Free

**Link:** https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/

**Tags:** complex-analysis, analytic-functions, residue-theorem, laurent-series, conformal-mapping

### 2. Introduction to Complex Analysis

**Author:** Petra Bonfert-Taylor

Wesleyan University course on complex analysis covering complex numbers, analytic functions, conformal mappings, Möbius transformations, contour integration, Cauchy's theorem, power and Laurent series, and residues. Learners finish able to evaluate contour integrals and analyze complex-differentiable functions.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.coursera.org/learn/complex-analysis

**Tags:** complex-analysis, analytic-functions, conformal-mapping, contour-integration, residue-theorem

## Books

### 1. Harmonic Function Theory

**Author:** Sheldon Axler, Paul Bourdon, Wade Ramey

Graduate Texts in Mathematics volume developing harmonic functions on Euclidean space: mean value and maximum principles, Poisson integrals, Harnack's inequality, bounded harmonic functions, harmonic polynomials, and the Dirichlet problem. Readers gain a self-contained command of classical harmonic function theory using only real analysis.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/3540978755?tag=edmonddante07-20

**Tags:** harmonic-functions, potential-theory, poisson-integral, dirichlet-problem, real-analysis

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