![Essential singularity](https://upload.wikimedia.org/wikipedia/commons/0/0b/Essential_singularity.png)
In
complex analysis, an essential singularity of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is a "severe"
singularity near which the function exhibits odd 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
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
s. In practice some include non-isolated singularities too; those do not have a
residue
Residue may refer to:
Chemistry and biology
* An amino acid, within a peptide chain
* Crop residue, materials left after agricultural processes
* Pesticide residue, refers to the pesticides that may remain on or in food after they are applied ...
.
Formal description
Consider an
open subset of the
complex plane . Let
be an element of
, and
a
holomorphic function. The point
is called an ''essential singularity'' of the function
if the singularity is neither a
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
nor a
removable singularity
In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbour ...
.
For example, the function
has an essential singularity at
.
Alternative descriptions
Let
be a complex number, assume that
is not defined at
but is
analytic in some region
of the complex plane, and that every
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of
has non-empty intersection with
.
:If both
and
exist, then
is a ''
removable singularity
In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbour ...
'' of both
and
.
: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
In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function.
Laurent series definition
The principal part at z=a of a function
: f(z) = \sum_^\infty a_ ...
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
.
On a
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
with a
point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
,
, 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. The
Riemann zeta function on the Riemann sphere has only one essential singularity, at
.
The behavior of
holomorphic functions near their essential singularities is described by the
Casorati–Weierstrass theorem
In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian ...
and by the considerably stronger
Picard's great theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
The theorems
Little Picard Theorem: If a function f: \mathbb \to\mathbb ...
. 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.)
References
*Lars V. Ahlfors; ''Complex Analysis'', McGraw-Hill, 1979
*Rajendra Kumar Jain, S. R. K. Iyengar; ''Advanced Engineering Mathematics''. Page 920. Alpha Science International, Limited, 2004.
{{refend
External links
* '
An Essential Singularity' by
Stephen Wolfram
Stephen Wolfram (; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer science, mathematics, and theoretical physics. In 2012, he was named a fellow of the American Ma ...
,
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
Essential Singularity on Planet Math
Complex analysis