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essential singularity[i′sen·chəl siŋ·gyə′lar·əd·ē]
If a function is single-valued and analytic in some neighborhood of the point z0 in the complex plane and there exists neither a finite nor an infinite limit for the function as z → z0, then z0 is said to be an essential singularity of the function (seeANALYTIC FUNCTIONS). For example, the point z = 0 is an essential singularity of such function as e1/z, z sin (1/z), and cos (1/z) + 1n (z + 1).
In a neighborhood of an essential singularity z0, the function f(z) can be expanded in a Laurent series:
Here, infinitely many of the numbers b1, b2, ... are nonzero. This property is often used to identify essential singularities.
The behavior of a function in the neighborhood of an essential singularity can be dealt with on the basis of the Casorati-Weier-strass theorem. A generalization of this theorem is provided by Picard’s big, or second, theorem: in every neighborhood of an essential singularity of an analytic function the function takes on every value, with at most one exception. Picard’s theorem has a number of extensions and refinements.
In some branches of the theory of analytic functions, the term “essential singularity” is also applied to singularities of a more complex nature.
REFERENCESMarkushevich, A. I. Teoriia analiticheskikh funktsii, 2nd ed., vols. 1–2. Moscow, 1967–68.
Nevanlinna, R. Odnoznachnye analiticheskie funktsii. Moscow-Leningrad, 1941. (Translated from German.)