In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, an essential singularity of a
function 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
poles. In practice some include non-isolated singularities too; those do not have a
residue.
Formal description
Consider an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. Let
be an element of
, and
a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
. The point
is called an ''essential singularity'' of the function
if the singularity is neither a
pole nor a
removable singularity.
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
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
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), ''Open'' (Blues Image album), 1969
* Open (Gotthard album), ''Open'' (Gotthard album), 1999
* Open (C ...
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
of
has non-empty intersection with
.
:If both
and
exist, then
is a ''
removable singularity'' 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
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
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. ...
,
, 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
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
on the Riemann sphere has only one essential singularity, at
.
The behavior of
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s 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.)
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,
Wolfram Demonstrations Project.
Essential Singularity on Planet Math
Complex analysis