Real and Imaginary Parts of Holomorphic Function are Harmonic

Theorem
Let $f: U \to \C$ be a holomorphic complex function on $U \subseteq \C$.

Let $u, v: D \to \C$ be real-valued functions defined as:

where:
 * $D = \set {\tuple {x, y}: x + i y \in U}$

Then $u$ and $v$ are harmonic functions.

Proof
By Cauchy-Riemann Equations, $u$ and $v$ satisfy:

Consider the partial derivative of $(1)$ $x$, and the partial derivative of $(2)$  $y$:

By Clairaut's Theorem:
 * $\dfrac {\partial^2 v} {\partial x \partial y} = \dfrac {\partial^2 v} {\partial y \partial x}$

Thus $(3) + (4)$ yields:
 * $\dfrac {\partial^2 u} {\partial x^2} + \dfrac {\partial^2 u} {\partial y^2} = 0$

So $u$ is a harmonic function by definition.

Now consider the partial derivative of $(1)$ $y$, and the partial derivative of $(2)$  $x$:

By Clairaut's Theorem :
 * $\dfrac {\partial^2 u} {\partial x \partial y} = \dfrac {\partial^2 u} {\partial y \partial x}$

Thus $(5) - (6)$ yields:
 * $0 = \dfrac {\partial^2 v} {\partial y^2} + \dfrac {\partial^2 v} {\partial x^2}$

So $v$ is a harmonic function by definition.