Cauchy-Riemann Equations

Theorem
Let $f: D \to \C$ be a complex function, where $D \subseteq \C$ is open.

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


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


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

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

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 $z \in D$:


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