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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the real function : f(x) = \frac has a singularity at x = 0, where the numerical value of the function approaches \pm\infty so the function is not defined. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, see singularity theory.


Real analysis

In real analysis, singularities are either discontinuities, or discontinuities of the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
(sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: type I, which has two subtypes, and type II, which can also be divided into two subtypes (though usually is not). To describe the way these two types of limits are being used, suppose that f(x) is a function of a real argument x, and for any value of its argument, say c, then the left-handed limit, f(c^-), and the right-handed limit, f(c^+), are defined by: :f(c^-) = \lim_f(x), constrained by x < c and :f(c^+) = \lim_f(x), constrained by x > c. The value f(c^-) is the value that the function f(x) tends towards as the value x approaches c from ''below'', and the value f(c^+) is the value that the function f(x) tends towards as the value x approaches c from ''above'', regardless of the actual value the function has at the point where x = c . There are some functions for which these limits do not exist at all. For example, the function :g(x) = \sin\left(\frac\right) does not tend towards anything as x approaches c = 0. The limits in this case are not infinite, but rather undefined: there is no value that g(x) settles in on. Borrowing from complex analysis, this is sometimes called an '' essential singularity''. The possible cases at a given value c for the argument are as follows. * A point of continuity is a value of c for which f(c^-) = f(c) = f(c^+), as one expects for a smooth function. All the values must be finite. If c is not a point of continuity, then a discontinuity occurs at c. * A type I discontinuity occurs when both f(c^-) and f(c^+) exist and are finite, but at least one of the following three conditions also applies: ** f(c^-) \neq f(c^+); ** f(x) is not defined for the case of x = c; or ** f(c) has a defined value, which, however, does not match the value of the two limits. *: *:Type I discontinuities can be further distinguished as being one of the following subtypes: ** A jump discontinuity occurs when f(c^-) \neq f(c^+), regardless of whether f(c) is defined, and regardless of its value if it is defined. ** A removable discontinuity occurs when f(c^-) = f(c^+), also regardless of whether f(c) is defined, and regardless of its value if it is defined (but which does not match that of the two limits). * A type II discontinuity occurs when either f(c^-) or f(c^+) 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. In other words, the function has an infinite discontinuity when its graph has a vertical asymptote. ** An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits f(c^-) or f(c^+) does not exist, but not because it is an ''infinite discontinuity''. ''Essential singularities'' approach no limit, not even if valid answers are extended to include \pm\infty. 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 singularities

A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this 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 (e.g., by replacing the latitude/longitude representation with an -vector representation).


Complex analysis

In complex analysis, there are several classes of singularities. These include the isolated singularities, the nonisolated singularities and the branch points.


Isolated singularities

Suppose that f is a function that is complex differentiable in the complement of a point a in an open subset U of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s \mathbb. Then: * The point a is a removable 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\setminus\. The function g is a continuous replacement for the function f. * The point a is a pole or non-essential singularity of f if there exists a holomorphic function g defined on U with g(a) nonzero, and a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
n such that f(z)=g(z)/(z-a)^n for all z in U\setminus\. The least such number n is called the ''order of the pole''. The derivative at a non-essential singularity itself has a non-essential singularity, with n increased by 1 (except if n is 0 so that the singularity is removable). * The point a is an essential singularity of f if it is neither a removable singularity nor a pole. The point a is an essential singularity
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
the Laurent series has infinitely many powers of negative degree.


Nonisolated singularities

Other than isolated singularities, complex functions of one variable may exhibit other singular behaviour. These are termed nonisolated singularities, of which there are two types: * Cluster points: limit points of isolated singularities. If they are all poles, despite admitting Laurent series expansions on each of them, then no such expansion is possible at its limit. * Natural boundaries: any non-isolated set (e.g. a curve) on which functions cannot be analytically continued around (or outside them if they are closed curves in the
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 ...
).


Branch points

Branch point In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, ...
s are generally the result of a multi-valued function, such as \sqrt or \log(z), which are 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 discontinuous 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 shape of the branch cut is a matter of choice, even though it must connect two different branch points (such as z = 0 and z = \infty for \log(z)) which are fixed in place.


Finite-time singularity

A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in
kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
s and Partial Differential Equations – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are power laws for various exponents of the form x^, of which the simplest is hyperbolic growth, where the exponent is (negative) 1: x^. More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses (t_0-t)^ (using ''t'' for time, reversing direction to -t so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time t_0). 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 In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
is lost on each bounce, the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
of bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of the
Painlevé paradox In rigid-body dynamics, the Painlevé paradox (also called frictional paroxysms by Jean Jacques Moreau) is the paradox that results from inconsistencies between the contact and Coulomb models of friction. It is named for former French prime minist ...
(for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a
coin A coin is a small, flat (usually depending on the country or value), round piece of metal or plastic used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order ...
spun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using the
Euler's Disk Euler's Disk, invented between 1987 and 1990 by Joseph Bendik, is a trademark for a scientific educational toy. It is used to illustrate and study the dynamic system of a spinning and rolling disk on a flat or curved surface. It has been the subje ...
toy). Hypothetical examples include
Heinz von Foerster Heinz von Foerster ( German spelling: Heinz von Förster; November 13, 1911 – October 2, 2002) was an Austrian American scientist combining physics and philosophy, and widely attributed as the originator of Second-order cybernetics. He was twice ...
's facetious " Doomsday's equation" (simplistic models yield infinite human population in finite time).


Algebraic geometry and commutative algebra

In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation defines a curve that has a cusp at the origin . One could define the -axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the -axis is a "double tangent." For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety. An equivalent definition in terms of
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
may be given, which extends to abstract varieties and schemes: A point is ''singular'' if the local ring at this point is not a regular local ring.


See also

* Catastrophe theory * Defined and undefined * Degeneracy (mathematics) * Division by zero * Hyperbolic growth * Pathological (mathematics) * Singular solution * Removable singularity


References

{{Authority control Mathematical analysis