holomorphic

Let $U\subset\mathbb{C}$ be a domain in the complex numbers. A function $f\colon U\longrightarrow\mathbb{C}$ is 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 holomorphic if $f$ is holomorphic in each of the variables. The class of all holomorphic functions on $\Omega$ is usually denoted by $\mathcal{O}(\Omega)$ .

Title	holomorphic
Canonical name	Holomorphic
Date of creation	2013-03-22 12:04:33
Last modified on	2013-03-22 12:04:33
Owner	djao (24)
Last modified by	djao (24)
Numerical id	12
Author	djao (24)
Entry type	Definition
Classification	msc 30D20
Classification	msc 32A10
Synonym	holomorphic function
Synonym	regular function
Synonym	complex differentiable
Related topic	CauchyRiemannEquations
Related topic	Analytic