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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク 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 $\backslash Gamma$ is a measure of the size of $\backslash Gamma$ that is invariant under conformal mappings. More specifically, suppose that $D$ is an open set in the complex plane and $\backslash Gamma$ is a collection of paths in $D$ and $f:D\backslash to\; D\text{'}$ is a conformal mapping. Then the extremal length of $\backslash Gamma$ is equal to the extremal length of the image of $\backslash Gamma$ under $f$. One also works with the conformal modulus of $\backslash Gamma$, the reciprocal of the extremal length. The fact that extremal length and conformal modulus are conformal invariants of $\backslash Gamma$ 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 $\backslash Gamma$ is a collection of rectifiable curves in $D$. If $\backslash rho:D\backslash to\; ()$ is Borel-measurable, then for any rectifiable curve $\backslash gamma$ we let :$L\_\backslash rho(\backslash gamma):=\backslash int\_\backslash gamma\; \backslash rho\backslash ,|dz|$ denote the $\backslash rho$-length of $\backslash gamma$, where $|dz|$ denotes the Euclidean element of length. (It is possible that $L\_\backslash rho(\backslash gamma)=\backslash infty$.) What does this really mean? If $\backslash gamma:I\backslash to\; D$ is parameterized in some interval $I$, then $\backslash int\_\backslash gamma\; \backslash rho\backslash ,|dz|$ is the integral of the Borel-measurable function $\backslash rho(\backslash gamma(t))$ with respect to the Borel measure on $I$ for which the measure of every subinterval $J\backslash subset\; I$ is the length of the restriction of $\backslash gamma$ to $J$. In other words, it is the Lebesgue-Stieltjes integral $\backslash int\_I\; \backslash rho(\backslash gamma(t))\backslash ,d\}\_\backslash gamma(t)$ is the length of the restriction of $\backslash gamma$ to $\backslash$. Also set :$L\_\backslash rho(\backslash Gamma):=\backslash inf\_L\_\backslash rho(\backslash gamma).$ The area of $\backslash rho$ is defined as :$A(\backslash rho):=\backslash int\_D\; \backslash rho^2\backslash ,dx\backslash ,dy,$ and the extremal length of $\backslash Gamma$ is :$EL(\backslash Gamma):=\; \backslash sup\_\backslash rho\; \backslash frac\backslash ,,$ where the supremum is over all Borel-measureable $\backslash rho:D\backslash to()$ with $\backslash Gamma\_0$ denotes the set of rectifiable curves in $\backslash Gamma$, then $EL(\backslash Gamma)$ is defined to be $EL(\backslash Gamma\_0)$. The term (conformal) modulus of $\backslash Gamma$ refers to $1/EL(\backslash Gamma)$. The extremal distance in $D$ between two sets in $\backslash overline\; D$ 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）』 ■ウィキペディアで「Extremal length」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.