Definition:Laplace's Equation/Complex Plane

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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:

\(\ds \map u {x, y}\) \(=\) \(\ds \map \Re {\map f z}\)
\(\ds \map v {x, y}\) \(=\) \(\ds \map \Im {\map f z}\)

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 is the second order PDE:

$\dfrac {\partial^2 u} {\partial x^2} + \dfrac {\partial^2 u} {\partial y^2} = \dfrac {\partial^2 v} {\partial x^2} + \dfrac {\partial^2 v} {\partial y^2} = 0$


Polar Form

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 known as

Laplace's equation is also known as the equation of continuity.

Some sources render it as (the) Laplace equation.


Also see

  • Results about Laplace's equation can be found here.


Source of Name

This entry was named for Pierre-Simon de Laplace.


Sources

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