Category:Cauchy-Riemann Equations

This category contains results about Cauchy-Riemann Equations.
Definitions specific to this category can be found in Definitions/Cauchy-Riemann Equations.

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:

$u$ and $v$ are differentiable in their entire domain.

Pages in category "Cauchy-Riemann Equations"

The following 5 pages are in this category, out of 5 total.