Definition:Cauchy-Riemann Equations

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Definition

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 $\map f z$.


The Cauchy-Riemann equations are the following equations:

\(\text {(1)}: \quad\) \(\ds \dfrac {\partial u} {\partial x}\) \(=\) \(\ds \dfrac {\partial v} {\partial y}\)
\(\text {(2)}: \quad\) \(\ds \dfrac {\partial u} {\partial y}\) \(=\) \(\ds -\dfrac {\partial v} {\partial x}\)

which hold for the partial derivatives of $u$ and $v$ if and only if:

$f$ is complex-differentiable in $D$
$u$ and $v$ are differentiable in their entire domain.


Also see

  • Results about the Cauchy-Riemann equations can be found here.


Source of Name

This entry was named for Augustin Louis Cauchy and Georg Friedrich Bernhard Riemann.


Historical Note

The Cauchy-Riemann Equations were used by Bernhard Riemann as the basis for his theory of the concept of the analytic function.


Sources

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