Complex derivative - Knowlpedia

Definition

Let $U\subseteq\mathbb C$ be open, $f:U\to\mathbb C$ , and $z_0\in U$ . The complex derivative of $f$ at $z_0$ is the limit

f'(z_0)=\lim_{\substack{h\to0\\h\in\mathbb C}}\frac{f(z_0+h)-f(z_0)}{h},

when it exists. The limit must have the same value as $h$ approaches $0$ through every complex direction.

Real-linear interpretation

Viewing $f$ as a map $\mathbb R^2\to\mathbb R^2$ , complex differentiability at $z_0$ is equivalent to real differentiability there with derivative equal to multiplication by one complex number. Thus its real derivative must commute with multiplication by $i$ . The Cauchy–Riemann equations express this constraint in coordinates.

Holomorphicity

A function is holomorphic on $U$ when it is complex differentiable at every point of $U$ . Differentiability at one point is much weaker: it need not imply continuity of the derivative or complex differentiability nearby.

References

Lars V. Ahlfors, Complex Analysis, 3rd ed., McGraw–Hill, 1979. Relevant: Chapter 2, §§1–2.