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 $u, v: \left\{ {\left({x, y}\right) \in \R^2: x + i y = z \in D }\right\} \to \R$ be two real-valued functions defined as:

$u \left({x, y}\right) = \operatorname{Re} \left({f \left({z}\right) }\right)$
$v \left({x, y}\right) = \operatorname{Im} \left({f \left({z}\right) }\right)$

where:

$\operatorname{Re} \left({f \left({z}\right)}\right)$ denotes the real part of $f \left({z}\right)$
$\operatorname{Im} \left({f \left({z}\right)}\right) $ denotes the imaginary part of $f \left({z}\right)$.


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 $v$:
$(1): \quad \dfrac {\partial u} {\partial x} = \dfrac {\partial v} {\partial y}$
$(2): \quad \dfrac {\partial u} {\partial y} = - \dfrac {\partial v} {\partial x}$


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

$f' \left({z}\right) = \dfrac{\partial f}{\partial x} \left({z}\right) = -i \dfrac{\partial f}{\partial y} \left({z}\right)$


Proof

Let $z = x + i y$.

Then:

\(\displaystyle \dfrac {\partial f} {\partial x} \left({z}\right)\) \(=\) \(\displaystyle \dfrac {\partial u} {\partial x} \left({x, y}\right) + i \dfrac {\partial v} {\partial x} \left({x, y}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \operatorname{Re} \left({f' \left({z}\right) }\right) + i \operatorname{Im} \left({f' \left({z}\right) }\right)\) from the last part of the proof for sufficient condition
\(\displaystyle \) \(=\) \(\displaystyle f' \left({z}\right)\)

Similarly:

\(\displaystyle -i \dfrac {\partial f} {\partial y} \left({z}\right)\) \(=\) \(\displaystyle -i \left({\dfrac {\partial u} {\partial y} \left({x, y}\right) + i \dfrac {\partial v} {\partial y} \left({x, y}\right) }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle -i \left({ -\operatorname{Im} \left({f' \left({z}\right) }\right) + i \operatorname{Re} \left({f' \left({z}\right) }\right) }\right)\) from the last part of the proof for sufficient condition
\(\displaystyle \) \(=\) \(\displaystyle f' \left({z}\right)\)

$\blacksquare$


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