About: Extremal length

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In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves is a measure of the size of that is invariant under conformal mappings. More specifically, suppose that is an open set in the complex plane and is a collectionof paths in and is a conformal mapping. Then the extremal length of is equal to the extremal length of the image of under . One also works with the conformal modulus of , the reciprocal of the extremal length. The fact that extremal length and conformal modulus are conformal invariants of 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

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  • In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves is a measure of the size of that is invariant under conformal mappings. More specifically, suppose that is an open set in the complex plane and is a collectionof paths in and is a conformal mapping. Then the extremal length of is equal to the extremal length of the image of under . One also works with the conformal modulus of , the reciprocal of the extremal length. The fact that extremal length and conformal modulus are conformal invariants of 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. (en)
  • 數學上，共形和擬共形映射的理論中，一個曲線族的極值長度是的一個共形不變量。確切來說，設是複平面中的開集，是中的路徑族，是一個共形映射。那麼的極值長度等於 在下的像的極值長度。因此極值長度是研究共形映射的有用工具。 (zh)
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  • 數學上，共形和擬共形映射的理論中，一個曲線族的極值長度是的一個共形不變量。確切來說，設是複平面中的開集，是中的路徑族，是一個共形映射。那麼的極值長度等於 在下的像的極值長度。因此極值長度是研究共形映射的有用工具。 (zh)
  • In the mathematical theory of conformal and quasiconformal mappings, the extremal length of a collection of curves is a measure of the size of that is invariant under conformal mappings. More specifically, suppose that is an open set in the complex plane and is a collectionof paths in and is a conformal mapping. Then the extremal length of is equal to the extremal length of the image of under . One also works with the conformal modulus of , the reciprocal of the extremal length. The fact that extremal length and conformal modulus are conformal invariants of 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 (en)
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  • Extremal length (en)
  • 極值長度 (zh)
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