Holomorphic Functions

Definition

Suppose $\Omega \subset \C $ is open and $f: \Omega \to \C $. Given $z_0 \in \C $, we say that $f$ is holomorphic at (or complex differentiable at) $z_0$ if the complex quotient

\[ \frac{f(z_0 + h) - f(z_0)}{h} \]

This condition is the complex analogue of a real function being differentiable at a point. The difference, of course, is that here $h$ is complex, and so the condition above encomposes all limits approaching $z$ (all angles) in the complex plane.¹ In other words, $h$ is a complex number approaching the complex number $(0, 0)$ from any direction. If the limit exists, then we call it the derivative of $f$ at $z_0$.

The function $f$ is holomorphic (regular, complex differentiable) on $\Omega $ if $f$ is holomorphic at every point of $\Omega $. If $C$ is a closed subset of $\C $, we say that $f$ is holomorphic on $C$ if $f$ is holomorphic on some open set containing $c$. If $f$ is holomorphic on all of $\C $ then we call $f$ entire.

Notation

In the case that $f: \Omega \to \C $ is holomorphic at $z_0$ we denote the derivative at $z_0$ by $f'(z_0)$. By definition,

\[ f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h} \]

1. Future editions will clarify. ↩︎