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 of an Analytic Function

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 $D$.

Worked Example

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

Cauchy-Riemann $v_y = u_x$:

Cauchy-Riemann $v_x = -u_y$: