Cauchy-Riemann Equations/Expression of Derivative

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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 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

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