Cauchy-Riemann Equations/Expression of Derivative

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 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 $\map f z$.

Then $f$ is complex-differentiable in $D$ if and only if:

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

and:

The following two equations, known as the Cauchy-Riemann equations, hold for the partial derivatives of $u$ and $v$:

$\text {(1)}: \quad$

$\ds \dfrac {\partial u} {\partial x}$

$=$

$\ds \dfrac {\partial v} {\partial y}$

$\text {(2)}: \quad$

$\ds \dfrac {\partial u} {\partial y}$

$=$

$\ds -\dfrac {\partial v} {\partial x}$

If the conditions are true, then for all $z \in D$:

$\map {f'} z = \map {\dfrac {\partial f} {\partial x} } z = -i \map {\dfrac {\partial f} {\partial y} } z$

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Proof

Let $z = x + i y$.

Then:

$\ds \map {\dfrac {\partial f} {\partial x} } z$

$=$

$\ds \map {\dfrac {\partial u} {\partial x} } {x, y} + i \map {\dfrac {\partial v} {\partial x} } {x, y}$

$\ds $

$=$

$\ds \map \Re {\map {f'} z} + i \, \map \Im {\map {f'} z}$

from the last part of the proof for sufficient condition

$\ds $

$=$

$\ds \map {f'} z$

Similarly:

$\ds -i \map {\dfrac {\partial f} {\partial y} } z$

$=$

$\ds -i \paren {\map {\dfrac {\partial u} {\partial y} } {x, y} + i \map {\dfrac {\partial v} {\partial y} } {x, y} }$

$\ds $

$=$

$\ds -i \paren {-\map \Im {\map {f'} z} + i \map \Re {\map {f'} z} }$

from the last part of the proof for sufficient condition

$\ds $

$=$

$\ds \map {f'} z$

$\blacksquare$

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Sources

2001: Christian Berg: Kompleks funktionsteori: $\S 1.3$