Essential singularity

essential singularitiessingularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.wikipedia
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Singularity (mathematics)

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

Complex analysis

complex variablecomplex functioncomplex functions
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.
Functions that have only poles but no essential singularities are called meromorphic.

Removable singularity

removable singularitiesremovableRiemann's theorem on removable singularities
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.

Zeros and poles

polepoleszero
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.
Thus a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function.

Casorati–Weierstrass theorem

Weierstrass–Casorati theoremWeierstrass-Casorati theorem
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.
In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities.

Principal part

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).
If the Laurent series has an inner radius of convergence of 0, then f(z) has an essential singularity at a, if and only if the principal part is an infinite sum.

Picard theorem

Picard's theoremPicard's great theoremPicard's little theorem
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.
Great Picard's Theorem: If an analytic function f has an essential singularity at a point w, then on any punctured neighborhood of w, f(z) takes on all possible complex values, with at most a single exception, infinitely often.

Laurent series

Laurent expansion theoremLaurent power seriesfield of Laurent series
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).
If the principal part of f is a finite sum, then f has a pole at c of order equal to (negative) the degree of the highest term; on the other hand, if f has an essential singularity at c, the principal part is an infinite sum (meaning it has infinitely many non-zero terms).

Open set

openopen subsetopen sets
Consider an open subset U of the complex plane \mathbb{C}. 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.

Complex plane

Argand diagramcomplex number planeArgand plane
Consider an open subset U of the complex plane \mathbb{C}.

Holomorphic function

holomorphicholomorphic mapholomorphic functions
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. Let a be an element of U, and a holomorphic function.

Analytic function

analyticanalytic functionsreal analytic
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.

Neighbourhood (mathematics)

neighbourhoodneighborhoodneighborhoods
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.

Émile Picard

Charles Émile PicardPicardEmile Picard
Picard's great theorem states that an analytic function with an essential singularity takes every value infinitely often, with perhaps one exception, in any neighborhood of the singularity.

Branch point

branch cutbranch pointsbranches
This is in contrast to transcendental and logarithmic branch points, that is, points at which a multiple-valued function has nontrivial monodromy and an essential singularity.

Residue (complex analysis)

residueresidueslogarithmic residue
(More generally, residues can be calculated for any function that is holomorphic except at the discrete points {a k } k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.

Value distribution theory of holomorphic functions

value distribution theoryNevanlinna theoryNevanlinna's theory
It tries to get quantitative measures of the number of times a function f(z) assumes a value a, as z grows in size, refining the Picard theorem on behaviour close to an essential singularity.

Isomonodromic deformation

isomonodromic deformationsFuchsian systemisomonodromy
This means that all essential singularities of the solutions are fixed, although the positions of poles may move.

Classification of discontinuities

discontinuitiesdiscontinuousdiscontinuity
(This is distinct from an essential singularity, which is often used when studying functions of complex variables.)

Felice Casorati (mathematician)

Felice CasoratiCasoratiCasorati, Felice
The theorem, named for Casorati and Karl Theodor Wilhelm Weierstrass, describes the remarkable behavior of holomorphic functions near essential singularities.

Entire function

entireHadamard productorder
Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function.