Essential Singularity

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 e^1/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.