Harmonic Functions

Let $D$ be a domain in the $xy$-plane. A real-valued function $u(x,y)$ is harmonic in $D$ iff:

• $u$ has continuous second-order partial derivatives in $D$.
• $u$ satisfies Laplace’s equation throughout $D$:

Component Functions

If $f(z) = u(x,y) + iv(x,y)$ is analytic in a domain $D$, then $u$ and $v$ are harmonic in $D$.

Harmonic Conjugate

Let $u$ and $v$ be real-valued functions on a region $D \subseteq \mathbb{R}^2$. $v$ is a harmonic conjugate of $u$ iff $f(z) = u(x,y) + iv(x,y)$ is analytic on $D$.

A harmonic conjugate of $u$ exists only if $u$ is harmonic on $D$. A harmonic $u$ does not always have one.

If $D$ is simply connected, a harmonic conjugate always exists and is unique up to an additive constant. Otherwise it may fail to exist.

For example: $u = \ln|z|$ is harmonic on $\mathbb{C} \setminus \{0\}$, but the only candidate $v = \arg z$ is multivalued, so no harmonic conjugate exists there.

Finding a Conjugate

Given a harmonic $u(x,y)$ on a simply connected domain:

1. Check that $u_{xx} + u_{yy} = 0$.
2. Compute $u_x$ and $u_y$.
3. Use the Cauchy-Riemann equation $v_y = u_x$. Integrate with respect to $y$ to get $v = \int u_x \, dy + \phi(x)$, where $\phi(x)$ is an unknown function of $x$ alone.
4. Differentiate this $v$ with respect to $x$ and equate it to $-u_y$, from the Cauchy-Riemann equation $v_x = -u_y$. All terms containing $y$ cancel because $u$ is harmonic, which leaves an expression for $\phi'(x)$.
5. Integrate $\phi'(x)$ to get $\phi(x)$, including an arbitrary constant $C$.
6. Substitute $\phi(x)$ into $v$. Then $f = u + iv$ is analytic, and can be written in terms of $z$.

Integrating $v_x = -u_y$ with respect to $x$ first and adding an unknown function of $y$ works the same way.

Worked Example

Find a harmonic conjugate $v$ of $u(x,y) = y^3 - 3x^2y$, and the corresponding analytic function $f$.