Essential singularity

Formal description

Consider an open subset U of the complex plane C. Let a be an element of U, and f : U  {a} → C a holomorphic function. The point a is called an essential singularity of the function f if the singularity is neither a pole nor a removable singularity.

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

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

lim z → a f ( z )   and   lim z → a 1 f ( z )   exist, then a is a removable singularity of both f and 1/f.

If

lim z → a f ( z )   exists but   lim z → a 1 f ( z )   does not exist, then a is a zero of f and a pole of 1/f.

Similarly, if

lim z → a f ( z )   does not exist but   lim z → a 1 f ( z )   exists, then a is a pole of f and a zero of 1/f.

If neither

lim z → a f ( z )   nor   lim z → a 1 f ( z )   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). A related definition is that if there is a point a for which f ( z ) ( z − a ) n is not differentiable for any integer n > 0 , then a is an essential singularity of f ( z ) .

The behavior of holomorphic functions near their 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.