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annulus (Definition)

An annulus is the region bounded between two (usually concentric) circles.

An open annulus is a domain in the complex plane of the form

$\displaystyle A = A_w(r,R) = \{z \in \mathbb{C}: r < \vert z-w\vert < R\}, $
where $ w$ is an arbitrary complex number, and $ r$ and $ R$ are real numbers with $ 0 < r < R$. Such a set is often called an annular region.

It should be noted that the word annulus usually refers to an open annulus.

More generally, one can allow $ r = 0$ or $ R = \infty$. (This makes sense for the purposes of the bound on $ \vert z-w\vert$ above.) This would make an annulus include the cases of a punctured disc, and some unbounded domains.

Analogously, a closed annulus is a set of the form

$\displaystyle \overline{A} = \overline{A}_w(r,R) = \{z \in \mathbb{C}: r \leq \vert z-w\vert \leq R\}, $
where $ w \in \mathbb{C}$, and $ r$ and $ R$ are real numbers with $ 0 < r < R$.

One can show that two annuli $ D_w(r,R)$ and $ D_{w'}(r',R')$ are conformally equivalent if and only if $ R/r = R'/r'$. More generally, the complement of any closed disk in an open disk is conformally equivalent to precisely one annulus of the form $ D_0(r,1)$.



"annulus" is owned by Wkbj79. [ full author list (2) | owner history (1) ]
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See Also: annulus

Other names:  open annulus, annular region
Also defines:  closed annulus
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Cross-references: open, closed, complement, conformally equivalent, unbounded, disc, bound, real numbers, complex number, complex plane, domain, circles, bounded, region
There are 9 references to this entry.

This is version 4 of annulus, born on 2003-04-22, modified 2007-06-26.
Object id is 4202, canonical name is Annulus2.
Accessed 5858 times total.

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