Cauchy-Riemann Equations

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:


 * $\map u {x, y} = \map \Re {\map f z}$


 * $\map v {x, y} = \map \Im {\map f z}$

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 $D$ :


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

and:
 * The following two equations, known as the Cauchy-Riemann equations, hold for the continuous 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 $z \in D$:


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