The Cauchy–Riemann Equations

In the setting of The Complex Plane as ℝ²: Standing Notation, let $f:U\to\mathbb{C}$ be a function, with $F$, $u$ and $v$ as in The Complex Plane as ℝ²: Standing Notation (iv), let $p_{0}=(x_{0},y_{0})\in V$, and let $z_{0}=\Phi(p_{0})=x_{0}+iy_{0}$. Complex differentiability and $f'(z_{0})$ are as in The Complex Derivative at a Point (ii), and differentiability of $F$ and derivative matrices as in Differentiability at a Point for Functions Between Euclidean Spaces (i) and Differentiability at a Point for Functions Between Euclidean Spaces (iii).

(i) The derivative matrix. Let $a,b\in\mathbb{R}$. Then $f$ is complex differentiable at $z_{0}$ with $f'(z_{0})=a+ib$ if and only if $F$ is differentiable at $p_{0}$ with derivative matrix

$$\begin{pmatrix}a&-b\\ b&a\end{pmatrix}.$$

(ii) The Cauchy–Riemann equations. $f$ is complex differentiable at $z_{0}$ if and only if $F$ is differentiable at $p_{0}$ and the partial derivatives of $u$ and $v$ at $p_{0}$ (which exist by The Derivative Is the Jacobian Matrix, and Is Unique (ii)) satisfy

$$\frac{\partial u}{\partial x}(p_{0})=\frac{\partial v}{\partial y}(p_{0}),\qquad\frac{\partial u}{\partial y}(p_{0})=-\frac{\partial v}{\partial x}(p_{0}).$$

(iii) The derivative. In that case

$$f'(z_{0})=\frac{\partial u}{\partial x}(p_{0})+i\,\frac{\partial v}{\partial x}(p_{0}).$$