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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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
Well-behaved

Mathematicians very frequently speak of whether a mathematics object — a number, a Function , a Set , a space of one sort or another — is "well-behaved" or not....
 in some particular way, such as differentiability
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
. See Singularity theory
Singularity theory

In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width....
 for general discussion of the geometric theory, which only covers some aspects.

For example, the function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....


on the real line
Real line

In mathematics, the real line is simply the set R of singleton real numbers.However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space....
 has a singularity at x = 0, where it seems to "explode" to ±8 and is not defined.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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
Well-behaved

Mathematicians very frequently speak of whether a mathematics object — a number, a Function , a Set , a space of one sort or another — is "well-behaved" or not....
 in some particular way, such as differentiability
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
. See Singularity theory
Singularity theory

In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width....
 for general discussion of the geometric theory, which only covers some aspects.

For example, the function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....


on the real line
Real line

In mathematics, the real line is simply the set R of singleton real numbers.However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space....
 has a singularity at x = 0, where it seems to "explode" to ±8 and is not defined. The function g(x) = |x| (see absolute value
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
) also has a singularity at x = 0, since it is not differentiable there. Similarly, the graph defined by y2 = x also has a singularity at (0,0), this time because it has a "corner" (vertical tangent) at that point.

The algebraic set defined by y2 = x2 in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 there.

Real analysis


In real analysis
Real analysis

Real analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the Set of real numbers. In particular, it deals with the analytic function properties of real function and sequences, including convergence and limit s of sequences of real numbers, the calculus of the real numbers, and continu...
 singularities are also called discontinuities
Classification of discontinuities

Continuous functions are of utmost importance in mathematics and applications. However, not all function are continuous. If a function is not continuous at a point in its domain , one says that it has a discontinuity there....
. There are three kinds: type I, which has two sub-types, and type II, which also can be divided into two subtypes, but normally is not.

To describe these types, suppose that is a function of a real argument , and for any value of its argument, say , the symbols and are defined by:

, constrained by and

, constrained by  .

The limit is called the left-handed limit, and is called the right-handed limit. The value is the value that the function tends towards as the value approaches from below, and the value is the value that the function tends towards as the value approaches from above, regardless of the actual value the function has at the point where  .

There are some functions for which these limits do not exist at all. For example the function does not tend towards anything as approaches . The limits in this case are not infinite, but rather undefined: there is no value that settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.

  • A point of continuity, which is not a singularity, is a value of for which , as one usually expects. All the values must be finite.
  • A type I discontinuity occurs when both and exist and are finite, but one of three conditions also apply: ; does not exist for that value of ; or does not match the value that the two limits tend towards. Two subtypes occur:
    • A jump discontinuity occurs when , regardless of whether exists, and regardless of what value it might have if it does exist.
    • A removable discontinuity
      Removable singularity

      In complex analysis, a removable singularity of a holomorphic function is a point at which the function is ostensibly undefined, but, upon closer examination, the domain of the function can be enlarged to include the mathematical singularity ....
       occurs when , but either the value of does not match the limits, or the function does not exist at the point  .
  • A type II discontinuity occurs when either or does not exist (possibly both). This has two subtypes, which are usually not considered separately:
    • An infinite discontinuity is the special case when either the left hand or right hand limit does not exist specifically because it is infinite, and the other limit is either also infinite or is some well defined finite number.
    • An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits or does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if legal answers are extended to include .


In real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, not to the original function.

Coordinate singularies


A coordinate singularity (or coördinate singularity) occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example is the apparent singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees). This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity.

Complex analysis


In complex analysis
Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating Function of complex numbers....
, there are four kinds of singularity, to be described below. Suppose U is an open subset
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
 of the complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s C, and the point a is an element of U, and f is a complex differentiable function
Holomorphic function

Holomorphic functions are the central object of study of complex analysis; they are function defined on an open set of the complex number C with values in C that are complex-differentiable at every point....
 defined on some neighborhood
Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a Set containing the point where you can move that point some amount without leaving the set....
 around a, excluding a: U \ .

  • Isolated singularities
    Isolated singularity

    In complex analysis, a branch of mathematics, an isolated singularity is one that has no other mathematical singularity close to it.Formally, a complex number z is an isolated singularity of a function f if there exists an open set disk D centered at z such that f is holomorphic function on D  , that is,...
    : Suppose the function f is not defined at a, although it does have values defined on U \ .
    • The point a is a removable singularity
      Removable singularity

      In complex analysis, a removable singularity of a holomorphic function is a point at which the function is ostensibly undefined, but, upon closer examination, the domain of the function can be enlarged to include the mathematical singularity ....
       of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U \ . The function g is a continuous replacement for the function f.
    • The point a is a pole
      Pole (complex analysis)

      In complex analysis, a mathematical discipline, a pole of a meromorphic function is a certain type of mathematical singularity that behaves like the singularity of at ....
       or non-essential singularity of f if there exists a holomorphic function g defined on U and a natural number
      Natural number

      In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
       n such that f(z) = g(z) / (z − a)n for all z in U \ . The derivative at a non-essential singularity may or may not exist. If g(a) is nonzero, then we say that a is a pole of order n
      Pole (complex analysis)

      In complex analysis, a mathematical discipline, a pole of a meromorphic function is a certain type of mathematical singularity that behaves like the singularity of at ....
      .
    • The point a is an essential singularity
      Essential singularity

      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...
       of f if is neither a removable singularity nor a pole. The point a is an essential singularity if and only if
      IFF

      IFF, Iff or iff can stand for:* Identification Friend or Foe, an electronic radio-based identification system utilizing transponders...
       the 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....
       has infinitely many powers of negative degree.
  • Branch point
    Branch point

    In the mathematics field of complex analysis, a branch point of a multivalued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point ....
    s are generally the result of a multi-valued function, such as or being defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontiuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The location and shape of most of the branch cut is usually a matter of choice, with perhaps only one point (like for ) which is fixed in place.


From the point of view of dynamics


A finite-time singularity occurs when a kinematic variable increases towards infinity at a finite time. An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy
Kinetic energy

The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the mechanical work needed to accelerate a body of a given mass from rest to its current velocity....
 is lost on each bounce, the frequency
Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
 of bounces becomes infinite as the ball comes to rest in a finite time. Other examples of finite-time singularities include Euler's disk
Euler's disk

Euler's disk, named after Leonhard Euler, is a circular disk that spins, without slipping, on a surface. The canonical example is a coin spinning on a table....
, the Painlevé paradox
Painlevé paradox

Painlev? paradox is a well known example by Paul Painlev? in rigid-body dynamics which, it was claimed, showed that rigid-body dynamics with contact and Coulomb friction is inconsistent....
, and Heinz von Foerster
Heinz von Foerster

Heinz von Foerster was an Austrian American scientist combining physics and philosophy. Together with Warren McCulloch, Norbert Wiener, John von Neumann, Lawrence J....
's Doomsday's Equation.

Algebraic geometry and commutative algebra


In algebraic geometry
Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry....
 and commutative algebra
Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideal , and module over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra....
, a singularity is a prime ideal
Prime ideal

In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. This article only covers ideals of ring theory....
 whose localization
Localization of a ring

In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring . Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of Unit in R*....
 is not a regular local ring
Regular local ring

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is exactly the same as its Krull dimension....
 (alternately a scheme
Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern algebraic geometry....
 with a stalk
Stalk (sheaf)

The stalk of a sheaf is a mathematics construction capturing the behaviour of a sheaf around a given point....
 that is not a regular local ring
Regular local ring

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is exactly the same as its Krull dimension....
). For example, defines an isolated singular point (at the cusp) . The ring in question is given by

The maximal ideal of the localization at is a height one local ring generated by two elements and thus not regular.

Singular matrices

In linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
 a square matrix is said to be singular when it is not invertible
Invertible matrix

In linear algebra, an n-by-n matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that...
, that is when its determinant
Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
 is zero, or equivalently when at least one of its eigenvalues is zero.

Singular value decomposition

Singular value decomposition
Singular value decomposition

In linear algebra, the singular value decomposition is an important Matrix decomposition of a rectangular real number or complex number matrix , with several applications in signal processing and statistics....
 (SVD) is a method of factorizing matrices. A non-negative real number s is a singular value for M if and only if there exist normalized vectors u in Km and v in Kn such that The vectors u and v are called left-singular and right-singular vectors for s, respectively. The factorisation is where diagonal entries of S are equal to the singular values of M. The columns of U and V are left- resp. right-singular vectors for the corresponding singular values. It is widely used in statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
 where it is used as a technique for solving linear least squares
Linear least squares

Linear least squares is an important computational problem, that arises primarily in applications when it is desired to fit a linear function mathematical model to measurements obtained from experiments....
 problems and is related to principal components analysis
Principal components analysis

Principal component analysis involves a mathematical procedure that transforms a number of possibly correlated variables into a smaller number of uncorrelated variables called principal components....
.

See also

  • Asymptote
    Asymptote

    An asymptote of a real-valued function is a curve which describes the behavior of as either or tends to infinity.In other words, as one moves along the graph of in some direction, the distance between it and the asymptote eventually becomes smaller than any distance that one may specify, and as the x or y values approach infinity, the...
  • Catastrophe theory
    Catastrophe theory

    In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry....
  • Discontinuity
    Discontinuity

    Discontinuity can be:*Discontinuity , a property of a mathematical functionDiscontinuity may also refer to:*A break in continuity , in literature...
  • Defined and undefined
    Defined and undefined

    In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion....
  • Hyperbolic growth
    Hyperbolic growth

    When a quantity grows towards a Mathematical singularity under a finite variation it is said to undergo hyperbolic growth.More precisely, the reciprocal function has a hyperbola as a graph, and has a singularity at 0, meaning that the limit as is infinity: any similar graph is said to exhibit hyperbolic growth....
  • Infinity
    Infinity

    Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
  • Singular solution
    Singular solution

    A singular solution ys of an ordinary differential equation differential equation is a solution that is tangent to every solution from the family of general solutions....
     of a differential equation
    Differential equation

    A differential equation is a mathematics equation for an unknown function of one or several variable that relates the values of the function itself and its derivatives of various orders....