---
tag: 031BG625Y3
kind: statement
role: proposition
name: The real part of a holomorphic function is harmonic
msc: [31A05]
uses: [0DG0Z72CQD, 0T71SSYZ39, 0QEBN8DGVE, 02ZE6JHJ7G]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_function
    locator: Connections with complex function theory
---

## Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.
Let $u = \operatorname{Re} f$ and $v = \operatorname{Im} f$, as real functions of $(x, y)$ on $U$, with $z = x + iy$.
Then $u$ and $v$ are twice continuously differentiable, and
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \qquad \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$$
on $U$. Thus $u$ and $v$ are harmonic functions on $U$.

## Remarks

Proof: $f$ is [[infinitely differentiable|0QEBN8DGVE]]. By the [[Cauchy–Riemann equations|0T71SSYZ39]], $u_{xx} = v_{yx} = v_{xy} = -u_{yy}$, and the same computation works for $v$.
For example, $\operatorname{Re}(z^2) = x^2 - y^2$ is harmonic, and so is $\operatorname{Re}(e^z) = e^x \cos y$.
The function $v$ is a [[harmonic conjugate|0CC56MJ4FG]] of $u$.

## Explanations

### L1 human

A harmonic function is a function of two real variables whose two pure second derivatives add up to $0$.
Each holomorphic function gives two harmonic functions for free: its real part and its imaginary part.