holomorphic Holomorphic 2001-12-28 05:15:28 2004-10-04 16:52:33 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} Let $U \subset \mathbb{C}$ be a domain in the complex numbers. A function $f \colon U \longrightarrow \mathbb{C}$ is \emph{holomorphic} if $f$ has a complex derivative at every point $x$ in $U$, i.e. if $$\lim_{z\rightarrow z_0} \frac{f(z)-f(z_0)}{z-z_0}$$ exists for all $z_0\in U$. More generally, if $\Omega\subset \mathbb{C}^n$ is a domain, then a function $f\colon \Omega \to \mathbb{C}$ is said to be \emph{holomorphic} if $f$ is holomorphic in each of the variables. The class of all holomorphic functions on $\Omega$ is usually denoted by $\mathcal{O}(\Omega)$.