In complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an isolated singularity is one that has no other singularities

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

 close to it. In other words, a complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 z₀ is an isolated singularity of a function f if there exists an open

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 disk

Disk (mathematics)

In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...

 D centered at z₀ such that f is holomorphic

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

 on D \ {z₀}, that is, on the set obtained from D by taking z₀ out.

Formally, and within the general scope of functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, an isolated singularity for a function is any topologically isolated point within an open set where the function is defined.

Every singularity of a meromorphic function

Meromorphic function

In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...

 is isolated, but isolation of singularities is not alone sufficient to guarantee a function is meromorphic. Many important tools of complex analysis such as Laurent series

Laurent series

In mathematics, the Laurent series of a complex function f 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...

 and the residue theorem

Residue theorem

The residue theorem, sometimes called Cauchy's Residue Theorem, in complex analysis is a powerful tool to evaluate line integrals of analytic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula...

 require that all relevant singularities of the function be isolated.

Examples

• The function has 0 as an isolated singularity.

• The cosecant function has every integer
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
   as an isolated singularity.

Nonisolated singularities

Other than isolated sigularities, complex functions of one variable may exhibit other singular behaviour. Namely, two kinds of nonisolated singularities exist:

• Cluster points, i.e. limit points of isolated singularities: if they are all poles, despite admitting Laurent series
  Laurent series
  In mathematics, the Laurent series of a complex function f 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...
  
   expansions on each of them, no such expansion is possible at its limit.

• Natural boundaries, i.e. any non-isolated set (e.g. a curve) which functions can not be analytically continued
  Analytic continuation
  In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
  
   around (or outside them if they are closed curves in the Riemann sphere
  Riemann sphere
  In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
  
  ).

Examples

• The function is meromorphic in , with simple poles in , for every . Since , every punctured disk centred in has an infinite number of singularities within, so no Laurent espansion is available for around , which is in fact a cluster point of its.

• The function has a singularity at 0 which is not isolated, since there are additional singularities at the reciprocal
  Multiplicative inverse
  In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
  
   of every integer
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
   which are located arbitrarily close to 0 (though the singularities at these reciprocals are themselves isolated).

• The function here defined as the Maclaurin series converges inside the unit circle centred in and has it boundarying circumference as natural boundary

External links

• Singularities Zeros, Poles by John H. Mathews