PlanetMath: free analytic boundary arc

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free analytic boundary arc

(Definition)

Definition 1 Let $G \subset \mathbb{C}$ be a region and let $\gamma$ be a connected subset of $\partial G$ (boundary of

), then $\gamma$ is a free analytic boundary arc of

if for every point $\zeta \in \gamma$ there is a neighbourhood

of $\zeta$ and a conformal equivalence $h \colon {\mathbb{D}} \to U$ (where ${\mathbb{D}}$ is the unit disc) such that $h(0) = \zeta$ , $h(-1,1) = \gamma \cap U$ and $h({\mathbb{D}}_+) = G \cap U$ (where ${\mathbb{D}}_+$ is all the points in the unit disc with non-negative imaginary part).

Bibliography

1: John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.

"free analytic boundary arc" is owned by jirka.

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See Also: analytic curve

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Cross-references: imaginary part, unit disc, conformal equivalence, neighbourhood, point, boundary, subset, connected, region
There is 1 reference to this entry.

This is version 4 of free analytic boundary arc, born on 2004-04-13, modified 2007-12-05.
Object id is 5758, canonical name is FreeAnalyticBoundaryArc.
Accessed 1239 times total.

Classification:

AMS MSC:	54-00 (General topology :: General reference works )
	30-00 (Functions of a complex variable :: General reference works )

Pending Errata and Addenda

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