Essential singularity

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

The category essential singularity is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles.

include non-isolated singularities too; those do not have a residue.

Consider an open subset

of the complex plane

a holomorphic function.

is called an essential singularity of the function

has an essential singularity at

be a complex number, and assume that

but is analytic in some region

of the complex plane, and that every open neighbourhood of

has non-empty intersection with

Another way to characterize an essential singularity is that the Laurent series of

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

converges to a limit as

is an essential singularity of

[1] On a Riemann sphere with a point at infinity,

has an essential singularity at that point if and only if the

has an essential singularity at 0: i.e. neither

[2] The Riemann zeta function on the Riemann sphere has only one essential singularity, at

[3] Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at

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

takes on every complex value, except possibly one, infinitely many times.

(The exception is necessary; for example, the function