Essential Singularity - Alternate Descriptions

Alternate Descriptions

Let a be a complex number, assume that f(z) is not defined at a but is analytic in some region U of the complex plane, and that every open neighbourhood of a has non-empty intersection with U.

If both

and exist, then a is a removable singularity of both f and 1/f.

If

exists but does not exist, then a is a zero of f and a pole of 1/f.

Similarly, if

does not exist but does exist, then a is a pole of f and a zero of 1/f.

If neither

nor exists, then a is an essential singularity of both f and 1/f.

Another way to characterize an essential singularity is that the Laurent series of f at the point a has infinitely many negative degree terms (i.e., the principal part of the Laurent series is an infinite sum).

The behavior of meromorphic functions near essential singularities is described by the Casorati–Weierstrass 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 many times.