Cauchy-Riemann Equations/Polar Form

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Theorem

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 can be expressed in polar form as:

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

where $z$ is expressed in exponential form as:

$z = r e^{i \theta}$


Proof


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Source of Name

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


Sources

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