Modulus of an annulus
The value reciprocal to the extremal length of the family of closed curves in the annulus which separate the boundary circles; the modulus is equal to
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By a conformal mapping onto an associated annulus , the modulus
of an annular domain
can be obtained. It turns out that
, where
is the Dirichlet integral of the real part of the function
mapping
onto
:
. (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 of a prime end (cf. Cluster set; Limit elements)
of an open Riemann surface
relative to a neighbourhood. Depending on whether
is finite or infinite, the prime end has hyperbolic or parabolic type and
either does or does not have a Green function.
For a simply-connected domain of hyperbolic type the so-called reduced modulus
relative to
is defined as the limit
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where is the modulus of the annular domain
. It turns out that
, where
is the conformal radius (cf. Conformal radius of a domain) of
relative to
.
Comments
References
[a1] | Z. Nehari, "Conformal mapping" , Dover, reprint (1975) |
Modulus of an annulus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modulus_of_an_annulus&oldid=47877