Harmonic Functions

One important concept often mentioned in multivariable calculus is the concept of a function being harmonic. We will define this concept below and later see its relevance in complex analysis. We first define a special function $\Delta (u)$ with respect to another function $u : A \subseteq \mathbb{R}^2 \to \mathbb{R}$ known as the Laplacian of $u$.

Definition: Let $A \subseteq \mathbb{R}^2$ be open and let $u : A \to \mathbb{R}$. If the second partial derivatives of $u$ with respect to the variables $x$ and $y$ exist on $A$, then the Laplacian of $u$ is defined to be the function $\displaystyle{ \Delta (u) = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}}$.

For example, consider the following function $u : \mathbb{R}^2 \to \mathbb{R}$:

(1)

\begin{align} \quad u(x, y) = x^2 + 2y^3 \end{align}

The partial derivatives of $u$ with respect to $x$ and $y$ are:

(2)

\begin{align} \quad \frac{\partial u}{\partial x} = 2x \quad \mathrm{and} \quad \frac{\partial u}{\partial y} = 6y^2 \end{align}

The second partial derivatives with respect to only the variables $x$ and $y$ are:

(3)

\begin{align} \quad \frac{\partial^2 u}{\partial x^2} = 2 \quad \mathrm{and} \quad \frac{\partial^2 u}{\partial y^2} = 12y \end{align}

These second partial derivatives exist on all of $\mathbb{R}^2$, so the Laplacian of $u$ is:

(4)

\begin{align} \quad \Delta (u) = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 2 + 12y \end{align}

When the Laplacian exists on a subset $A$ of $\mathbb{R}^2$ and equals to $0$ on all of $A$ then we give such a function a special name on $A$ which we define below.

Definition: Let $A \subseteq \mathbb{R}^2$ be open anad let $u : A \to \mathbb{R}$. If the second partial derivatives of $u$ with respect to the variables $x$ and $y$ exist on $A$ then $u$ is said to be Harmonic on $A$ if the Laplacian of $u$ equals $0$ on all of $A$, that is, $\displaystyle{ \Delta (u) = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0}$ on all of $A$.

For example, consider the function $u : \mathbb{R}^2 \to \mathbb{R}$ defined by:

(5)

\begin{align} \quad u(x, y) = x^2 - y^2 \end{align}

We claim that $u$ is harmonic on all of $\mathbb{R}^2$. The partial derivatives of $u$ with respect to $x$ and $y$ are:

(6)

\begin{align} \quad \frac{\partial u}{\partial x} = 2x \quad \mathrm{and} \quad \frac{\partial u}{\partial y} = -2y \end{align}

The second partial derivatives of $u$ with respect to only $x$ and only $y$ are:

(7)

\begin{align} \quad \frac{\partial^2 u}{\partial x^2} = 2 \quad \mathrm{and} \quad \frac{\partial^2 u}{\partial y^2} = -2 \end{align}

So the Laplacian of $u$ is given on all of $\mathbb{R}^2$ by:

(8)

\begin{align} \quad \Delta (u) = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 2 + (-2) = 0 \end{align}

So $u$ is harmonic on all of $\mathbb{R}^2$.

We will soon see the importance of the concept of harmonic functions in terms of analytic functions.