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Extremal length : ウィキペディア英語版
Extremal length
In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves \Gamma is a measure of the size of \Gamma that is invariant under conformal mappings. More specifically, suppose that D is an open set in the complex plane and \Gamma is a collection
of paths in D and f:D\to D' is a conformal mapping. Then the extremal length of \Gamma is equal to the extremal length of the image of \Gamma under f. One also works with the conformal modulus of \Gamma, the reciprocal of the extremal length. The fact that extremal length and conformal modulus are conformal invariants of \Gamma makes them useful tools in the study of conformal and quasi-conformal mappings. One also works with extremal length in dimensions greater than two and certain other metric spaces, but the following deals primarily with the two dimensional setting.
==Definition of extremal length==
To define extremal length, we need to first introduce several related quantities.
Let D be an open set in the complex plane. Suppose that \Gamma is a
collection of rectifiable curves in D. If \rho:D\to ()
is Borel-measurable, then for any rectifiable curve \gamma we let
:L_\rho(\gamma):=\int_\gamma \rho\,|dz|
denote the \rho-length of \gamma, where |dz| denotes the
Euclidean element of length. (It is possible that L_\rho(\gamma)=\infty.)
What does this really mean?
If \gamma:I\to D is parameterized in some interval I,
then \int_\gamma \rho\,|dz| is the integral of the Borel-measurable function
\rho(\gamma(t)) with respect to the Borel measure on I
for which the measure of every subinterval J\subset I is the length of the
restriction of \gamma to J. In other words, it is the
Lebesgue-Stieltjes integral
\int_I \rho(\gamma(t))\,d}_\gamma(t) is the length of the restriction of \gamma
to \.
Also set
:L_\rho(\Gamma):=\inf_L_\rho(\gamma).
The area of \rho is defined as
:A(\rho):=\int_D \rho^2\,dx\,dy,
and the extremal length of \Gamma is
:EL(\Gamma):= \sup_\rho \frac\,,
where the supremum is over all Borel-measureable \rho:D\to() with 0. If \Gamma contains some non-rectifiable curves and
\Gamma_0 denotes the set of rectifiable curves in \Gamma, then
EL(\Gamma) is defined to be EL(\Gamma_0).
The term (conformal) modulus of \Gamma refers to 1/EL(\Gamma).
The extremal distance in D between two sets in \overline D is the extremal length of the collection of curves in D with one endpoint in one set and the other endpoint in the other set.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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