Plot of the function exp(1/z), centered on the essential singularity at z = 0. The hue represents the
complex argument, the luminance represents the
absolute value. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white).Model illustrating essential singularity of a complex function 6w = exp(1/(6z))
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. In practice some[who?] include non-isolated singularities too; those do not have a
residue.
For example, the function has an essential singularity at .
Alternative descriptions
Let be a
complex number, and assume that is not defined at but is
analytic in some region of the complex plane, and that every
openneighbourhood of has non-empty intersection with .
If exists but does not exist (in fact ), then is a
zero of and a
pole of .
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of .
If neither nor exists, then is an essential singularity of both and .
Another way to characterize an essential singularity is that the
Laurent series of at the point 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 for which no derivative of converges to a limit as tends to , then is an essential singularity of .[1]
On a
Riemann sphere with a
point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i.e. neither nor exists.[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 , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)