Essential singularity

related topics
{math, number, function}
{math, energy, light}

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

Basically, the category essential singularity is a "left-over" or default group of 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.

Contents

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 meromorphic 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) = e1/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

If

Similarly, if

If neither

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 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 many times.

References

Full article ▸

related documents
Direct sum of groups
Derivative of a constant
Greibach normal form
Sigmoid function
Hilbert's Nullstellensatz
Rectangle
Constant folding
Linearity of integration
Recursive language
Z notation
List of Fourier-related transforms
Lazy initialization
Discrete mathematics
Distinct
Euler's theorem
Group object
Galois group
Context-sensitive language
Product of group subsets
Disjoint sets
Identity function
Harmonic analysis
Surjective function
Context-free language
Online algorithm
Sedenion
CycL
The Third Manifesto
Water, gas, and electricity
Markov process