03 - Holomorphic Implies Harmonic

Holomorphic Implies Harmonic

Let $U\subseteq \mathbb{C}$ be open, and suppose that $f:U\to \mathbb{C}$ is holomorphic
Let components of $f$ be $(u,v)$, and suppose both are twice continuously differentiable then

$$u\text{ and }v\text{ are harmonic}$$

Proof

From the Holomorphic Implies Harmonic then

$${\partial }_{xx}u={\partial }_{xy}v(={\partial }_x{\partial }_{y}v),{\partial }_{yy}u=-{\partial }_{yx}v$$

However under the stated conditions, the symmetry property of partial derivatives apply

$${\partial }_{xy}v={\partial }_{yx}v$$

Hence the result follows