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单词 EssentialSingularity
释义

essential singularity


Let U be a domain, aU, and let f:U{a} be holomorphic. If the Laurent seriesMathworldPlanetmath expansion of f(z) around a contains infinitely many terms with negative powers of z-a, then a is said to be an essential singularityMathworldPlanetmath of f. Any singularity of f is a removable singularityMathworldPlanetmath, a pole or an essential singularity.

If a is an essential singularity of f, then the image of any punctured neighborhoodMathworldPlanetmath of a under f is dense in (the Casorati-Weierstrass theorem). In fact, an even stronger statement is true: according to Picard’s theorem, the image of any punctured neighborhood of a is , with the possible exception of a single point.

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更新时间:2025/5/4 19:06:43