# 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 $u, v: \left\{ {\left({x, y}\right) \in \R^2: x + i y = z \in D }\right\} \to \R$ 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 $\displaystyle$ $=$ $\displaystyle f' \left({z}\right)$

$\blacksquare$