PlanetMath: holomorphic

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holomorphic

(Definition)

Let $U \subset \mathbb{C}$ be a domain in the complex numbers. A function $f \colon U \longrightarrow \mathbb{C}$ is holomorphic if has a complex derivative at every point in , i.e. if

$\displaystyle \lim_{z\rightarrow z_0} \frac{f(z)-f(z_0)}{z-z_0}$

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 is holomorphic in each of the variables. The class of all holomorphic functions on $\Omega$ is usually denoted by $\mathcal{O}(\Omega)$ .

"holomorphic" is owned by djao. [ full author list (2) | owner history (1) ]

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See Also: Cauchy-Riemann equations, analytic

Other names:

holomorphic function, regular function, complex differentiable

Attachments:: holomorphic functions of several variables (Definition) by jirka
identity theorem of holomorphic functions (Theorem) by rspuzio
regular at infinity (Definition) by pahio

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Cross-references: class, variables, point, complex derivative, function, complex numbers, domain
There are 136 references to this entry.

This is version 8 of holomorphic, born on 2001-12-28, modified 2004-10-04.
Object id is 1146, canonical name is Holomorphic.
Accessed 18705 times total.

Classification:

AMS MSC:	30D20 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Entire functions, general theory)
	32A10 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Holomorphic functions)

Pending Errata and Addenda

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