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'annulus'
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| Title of object: |
annulus |
| Canonical Name: |
Annulus2 |
| Type: |
Definition |
| Created on: |
2003-04-22 23:04:55 |
| Modified on: |
2003-04-22 23:04:55 |
| Classification: |
msc:30A99 |
| Synonyms: |
annulus=annular region |
Revision comment (for changes between this and next version):
| Changes for correction #11922 ('typo'). |
Preamble:
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\newcommand{\bbC}{\mathbb{C}}
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Content:
Briefly, an annulus is the region bounded between two (usually concentric) circles.
An open annulus, or just annulus for short, is a domain in the complex plane of the form
\[
A = A_w(r,R) = \{z \in \bbC \mid r < |z-w| < R\},
\]
where $w$ is an abitrary complex number, and $r$ and $R$ are real numbers with $0 < r < R$. Such a set is often called an annular region.
More generally, one can allow $r = 0$ or $R = \infty$. (This makes sense for the purposes of the bound on $|z-w|$ above.) This would make an annulus include the cases of a punctured disc, and some unbounded domains.
Analogously, one can define a closed annulus to be a set of the form
\[
\ov{A} = \ov{A}_w(r,R) = \{z \in \bbC \mid r \leq |z-w| \leq R\},
\]
where $w \in \bbC$, 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 equivalen to precisely one annulus of the form $D_0(r,1)$. |
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