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# 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 $\mathcal{O}(\Omega)$.

Related:

CauchyRiemannEquations, Analytic

Synonym:

holomorphic function, regular function, complex differentiable

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

30D20*no label found*32A10

*no label found*

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## Corrections

msc/domain by nerdy2 ✓

synonyms by Daume ✓

More general by bbukh ✓

defined also.. by matte ✓

images distorted by cxseven ✘

synonyms by Daume ✓

More general by bbukh ✓

defined also.. by matte ✓

images distorted by cxseven ✘