Definition:Laplace's Equation/Complex Plane/Polar Form

Equation
Let $D \subseteq \C$ be an open subset of the set of complex numbers $\C$.

Let $f: D \to \C$ be a complex function on $D$.

Let $u, v: \set {\tuple {x, y} \in \R^2: x + i y = z \in D} \to \R$ be the two real-valued functions defined as:

where:
 * $\map \Re {\map f z}$ denotes the real part of $\map f z$
 * $\map \Im {\map f z}$ denotes the imaginary part of $\map f z$.

Laplace's equation can be expressed in the form:


 * $r \map {\dfrac \partial {\partial r} } {r \dfrac {\partial u} {\partial r} } + \dfrac {\partial^2 u} {\partial \theta^2} = r \map {\dfrac \partial {\partial r} } {r \dfrac {\partial v} {\partial r} } + \dfrac {\partial^2 v} {\partial \theta^2} = 0$

where $z$ is expressed in exponential form as:
 * $z = r e^{i \theta}$

Also see

 * Solution to Laplace's Equation