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:

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 $D$ :


 * $u$ 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 $v$:

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


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

Proof
Let $z = x + i y$.

Then:

Similarly: