# 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