Cauchy-Riemann Equations/Polar Form

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

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Complex Functions, Cauchy-Riemann Equations: $3.7.31$