Essential singularity - Definition and Overview

In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.

Formally, consider an open subset U of the complex plane C, an element a of U, and a holomorphic function f defined on U - {a}. The point a is called an essential singularity for f if it is a singularity which is neither a pole nor a removable singularity.

For example, the function f(z) = exp(1/z) has an essential singularity at a = 0.

The point a is an essential singularity if and only if the limit

<math>\lim_{z\to a}f(z)<math>

does not exist as a complex number nor equals infinity. This is the case if and only if the Laurent series of f at the point a has infinitely many negative degree terms (the principal part is an infinite sum).

The behavior of holomorphic functions near essential singularities is described by the Weierstrass-Casorati theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity a, the function f takes on every complex value, except possibly one, infinitely often.