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The value reciprocal to the [[Extremal length|extremal length]] of the family of closed curves in the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m0645401.png" /> which separate the boundary circles; the modulus is equal to
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m0645402.png" /></td> </tr></table>
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By a [[Conformal mapping|conformal mapping]] onto an associated annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m0645403.png" />, the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m0645404.png" /> of an annular domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m0645405.png" /> can be obtained. It turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m0645406.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m0645407.png" /> is the [[Dirichlet integral|Dirichlet integral]] of the real part of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m0645408.png" /> mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m0645409.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454010.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454011.png" />. (Thus, a given [[Annular domain|annular domain]] is mapped onto an annulus with a fixed ratio of the radii of the boundary circles. This fact can be taken as another definition of the modulus of an annulus, its generalization leads to the idea of the modulus of a plane domain.)
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The value reciprocal to the [[Extremal length|extremal length]] of the family of closed curves in the annulus  $  r _ {1} \leq  | z | \leq  r _ {2} $
 +
which separate the boundary circles; the modulus is equal to
  
A generalization of the modulus of an annular domain is the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454012.png" /> of a prime end (cf. [[Cluster set|Cluster set]]; [[Limit elements|Limit elements]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454013.png" /> of an open [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454014.png" /> relative to a neighbourhood. Depending on whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454015.png" /> is finite or infinite, the prime end has hyperbolic or parabolic type and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454016.png" /> either does or does not have a [[Green function|Green function]].
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$$
  
For a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454017.png" /> of hyperbolic type the so-called reduced modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454018.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454019.png" /> is defined as the limit
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\frac{1}{2 \pi }
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  \mathop{\rm ln} 
 +
\frac{r _ {2} }{r _ {1} }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454020.png" /></td> </tr></table>
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By a [[Conformal mapping|conformal mapping]] onto an associated annulus  $  K $,
 +
the modulus  $  m _ {G} $
 +
of an annular domain  $  G $
 +
can be obtained. It turns out that  $  m _ {G} = D ( u) / 2 \pi $,
 +
where  $  D ( u) $
 +
is the [[Dirichlet integral|Dirichlet integral]] of the real part of the function  $  u $
 +
mapping  $  G $
 +
onto  $  K $:  
 +
$  \{ 1 < | w | < e ^ {m _ {G} } \} $.  
 +
(Thus, a given [[Annular domain|annular domain]] is mapped onto an annulus with a fixed ratio of the radii of the boundary circles. This fact can be taken as another definition of the modulus of an annulus, its generalization leads to the idea of the modulus of a plane domain.)
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454021.png" /> is the modulus of the annular domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454022.png" />. It turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454024.png" /> is the conformal radius (cf. [[Conformal radius of a domain|Conformal radius of a domain]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454025.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064540/m06454026.png" />.
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A generalization of the modulus of an annular domain is the modulus  $  m _  \gamma  $
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of a prime end (cf. [[Cluster set|Cluster set]]; [[Limit elements|Limit elements]]) $  \gamma $
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of an open [[Riemann surface|Riemann surface]]  $  R $
 +
relative to a neighbourhood. Depending on whether  $  m _  \gamma  $
 +
is finite or infinite, the prime end has hyperbolic or parabolic type and  $  R $
 +
either does or does not have a [[Green function|Green function]].
  
 +
For a simply-connected domain  $  D $
 +
of hyperbolic type the so-called reduced modulus  $  m _ {z _ {0}  } $
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relative to  $  z _ {0} \in D $
 +
is defined as the limit
  
 +
$$
 +
\lim\limits _ {r \rightarrow 0 }
 +
\left (
 +
m ( r) +
 +
\frac{1}{2 \pi }
 +
  \mathop{\rm ln}  r
 +
\right ) ,
 +
$$
 +
 +
where  $  m ( r) $
 +
is the modulus of the annular domain  $  D ( r) = D \cap \{ | z - z _ {0} | > r \} $.
 +
It turns out that  $  m _ {z _ {0}  } = (  \mathop{\rm ln}  R ) / 2 \pi $,
 +
where  $  R $
 +
is the conformal radius (cf. [[Conformal radius of a domain|Conformal radius of a domain]]) of  $  D $
 +
relative to  $  z _ {0} \in D $.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Z. Nehari,  "Conformal mapping" , Dover, reprint  (1975)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Z. Nehari,  "Conformal mapping" , Dover, reprint  (1975)</TD></TR></table>

Latest revision as of 08:01, 6 June 2020


The value reciprocal to the extremal length of the family of closed curves in the annulus $ r _ {1} \leq | z | \leq r _ {2} $ which separate the boundary circles; the modulus is equal to

$$ \frac{1}{2 \pi } \mathop{\rm ln} \frac{r _ {2} }{r _ {1} } . $$

By a conformal mapping onto an associated annulus $ K $, the modulus $ m _ {G} $ of an annular domain $ G $ can be obtained. It turns out that $ m _ {G} = D ( u) / 2 \pi $, where $ D ( u) $ is the Dirichlet integral of the real part of the function $ u $ mapping $ G $ onto $ K $: $ \{ 1 < | w | < e ^ {m _ {G} } \} $. (Thus, a given annular domain is mapped onto an annulus with a fixed ratio of the radii of the boundary circles. This fact can be taken as another definition of the modulus of an annulus, its generalization leads to the idea of the modulus of a plane domain.)

A generalization of the modulus of an annular domain is the modulus $ m _ \gamma $ of a prime end (cf. Cluster set; Limit elements) $ \gamma $ of an open Riemann surface $ R $ relative to a neighbourhood. Depending on whether $ m _ \gamma $ is finite or infinite, the prime end has hyperbolic or parabolic type and $ R $ either does or does not have a Green function.

For a simply-connected domain $ D $ of hyperbolic type the so-called reduced modulus $ m _ {z _ {0} } $ relative to $ z _ {0} \in D $ is defined as the limit

$$ \lim\limits _ {r \rightarrow 0 } \left ( m ( r) + \frac{1}{2 \pi } \mathop{\rm ln} r \right ) , $$

where $ m ( r) $ is the modulus of the annular domain $ D ( r) = D \cap \{ | z - z _ {0} | > r \} $. It turns out that $ m _ {z _ {0} } = ( \mathop{\rm ln} R ) / 2 \pi $, where $ R $ is the conformal radius (cf. Conformal radius of a domain) of $ D $ relative to $ z _ {0} \in D $.

Comments

References

[a1] Z. Nehari, "Conformal mapping" , Dover, reprint (1975)
How to Cite This Entry:
Modulus of an annulus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modulus_of_an_annulus&oldid=47877
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article