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| 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 (mathematics)|set]] where it fails to be [[well-behaved]] in some particular way, such as [[derivative|differentiability]]. See [[Singularity theory]] for general discussion of the [[geometric]] theory, which only covers some aspects.
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| For example, the [[function (mathematics)|function]]
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| :<math> f(x)=\frac{1}{x} </math>
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| on the [[real line]] has a singularity at ''x'' = 0, where it seems to "explode" to ±∞ and is not defined. The function ''g''(''x'') = |''x''| (see [[absolute value]]) also has a singularity at ''x'' = 0, since it is not differentiable there. Similarly, the graph defined by ''y''<sup>2</sup> = ''x'' also has a singularity at (0,0), this time because it has a "corner" (vertical tangent) at that point.
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| The algebraic set defined by <math>\{(x,y):|x|=|y|\}</math> in the (''x'', ''y'') coordinate system has a singularity (singular point) at (0, 0) because it does not admit a [[tangent]] there.
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| ==Real analysis==
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| In [[real analysis]] singularities are either [[classification of discontinuities|discontinuities]] or discontinuities of the [[derivative]] (sometimes also discontinuities of higher order derivatives). There are three kinds of discontinuities: '''type I''', which has two sub-types, and '''type II''', which also can be divided into two subtypes, but normally is not.
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| To describe these types, suppose that <math>f(x)</math> is a function of a real argument <math>x</math>, and for any value of its argument, say <math>c</math>, the symbols <math>f(c^+)</math> and <math>f(c^-)</math> are defined by:
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| :<math>f(c^+) = \lim_{x \to c}f(x)</math>, constrained by <math>x > c\ </math> and
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| :<math>f(c^-) = \lim_{x \to c}f(x)</math>, constrained by <math>x < c\ </math> .
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| The [[limit of a function|limit]] <math>f(c^-)</math> is called the '''left-handed limit''', and <math>f(c^+)</math> is called the '''right-handed limit'''. The value <math>f(c^-)</math> is the value that the function <math>f(x)</math> ''tends towards'' as the value <math>x</math> approaches <math>c</math> from below, and the value <math>f(c^+)</math> is the value that the function <math>f(x)</math> ''tends towards'' as the value <math>x</math> approaches <math>c</math> from above, regardless of the actual value the function has at the point where <math>x = c</math> .
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| There are some functions for which these limits do not exist at all. For example the function
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| :<math>g(x) = \sin\left(\frac{1}{x}\right)</math>
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| does not tend towards anything as <math>x</math> approaches <math>c = 0</math>. The limits in this case are not infinite, but rather undefined: there is no value that <math>g(x)</math> settles in on. Borrowing from complex analysis, this is sometimes called an ''essential singularity''.
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| * A '''point of continuity''', is a value of <math>c</math> for which <math>f(c^-) = f(c) = f(c^+)</math>, as one usually expects. All the values must be finite.
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| * A '''type I''' discontinuity occurs when both <math>f(c^-)</math> and <math>f(c^+)</math> exist and are finite, but one of three conditions also apply: <math>f(c^-) \neq f(c^+)</math>; <math>f(c)</math> does not exist for that value of <math>x</math>; or <math>f(c)</math> does not match the value that the two limits tend towards. Two subtypes occur:
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| ** A '''[[jump discontinuity]]''' occurs when <math>f(c^-) \neq f(c^+)</math>, regardless of whether <math>f(c)</math> exists, and regardless of what value it might have if it does exist.
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| ** A '''[[removable singularity|removable discontinuity]]''' occurs when <math>f(c^-) = f(c^+)</math>, but either the value of <math>f(c)</math> does not match the limits, or the function does not exist at the point <math>x = c</math> .
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| * A '''type II''' discontinuity occurs when either <math>f(c^-)</math> or <math>f(c^+)</math> does not exist (possibly both). This has two subtypes, which are usually not considered separately:
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| ** 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.
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| ** An '''essential singularity''' is a term borrowed from complex analysis (see below). This is the case when either one or the other limits <math>f(c^-)</math> or <math>f(c^+)</math> 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 <math>\pm\infty</math>.
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| 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.
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| ===Coordinate singularities===
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| {{Main|Coordinate singularity}}
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| 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, e.g. by replacing latitude/longitude with [[n-vector]].
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| ==Complex analysis==
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| In [[complex analysis]] there are four classes of singularities, described below. Suppose ''U'' is an [[open set|open subset]] of the [[complex number]]s '''C''', and the point ''a'' is an element of ''U'', and ''f'' is a [[holomorphic function|complex differentiable function]] defined on some [[Neighbourhood (mathematics)|neighborhood]] around ''a'', excluding ''a'': ''U'' \ {''a''}.
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| * [[Isolated singularity|Isolated singularities]]: Suppose the function ''f'' is not defined at ''a'', although it does have values defined on ''U'' \ {''a''}.
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| ** 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'' \ {''a''}. The function ''g'' is a continuous replacement for the function ''f''.
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| ** The point ''a'' is a [[pole (complex analysis)|pole]] or non-essential singularity of ''f'' if there exists a holomorphic function ''g'' defined on ''U'' and a [[natural number]] ''n'' such that ''f''(''z'') = ''g''(''z'') / (''z'' − ''a'')<sup>''n''</sup> for all ''z'' in ''U'' \ {''a''}. 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 (complex analysis)|pole of order ''n'']].
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| ** 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 [[iff|if and only if]] the [[Laurent series]] has infinitely many powers of negative degree.
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| * [[Branch point]]s are generally the result of a [[multi-valued function]], such as <math>\sqrt{z}</math> or <math>\log(z)</math> 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 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, however, it must connect two different branch points (like <math>z=0</math> and <math>z=\infty</math> for <math>\log(z)</math>) which is fixed in place.
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| ==Finite-time singularity==
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| [[Image:Rectangular hyperbola.svg|thumb|The [[reciprocal function]], exhibiting [[hyperbolic growth]].]]<!-- A better image would be 1/(1-x) or similar, showing a positive singular point and growth as x increases -->
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| A '''finite-time singularity''' occurs when one input variable is time, and an output variable increases towards infinite at a finite time. These are important in [[kinematic]]s and PDEs – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically the simplest finite-time singularities are [[power law]]s for various exponents, <math>x^{-\alpha},</math> of which the simplest is [[hyperbolic growth]], where the exponent is (negative) 1: <math>x^{-1}.</math> More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses <math>(t_0-t)^{-\alpha}</math> (using ''t'' for time, reversing direction to <math>-t</math> so time increases to infinity, and shifting the singularity forward from 0 to a fixed time <math>t_0</math>).
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| 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]] is lost on each bounce, the [[frequency]] of bounces becomes infinite as the ball comes to rest in a finite time. Other examples of finite-time singularities include the [[Painlevé paradox]] in various forms (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a [[coin]] spun on a flat surface accelerates towards infinite, before abruptly stopping (as studied using the [[Euler's Disk]] toy).
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| Hypothetical examples include [[Heinz von Foerster]]'s facetious "[[Heinz von Foerster#Doomsday Equation|Doomsday's Equation]]" (simplistic models yield infinite human population in finite time).
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| ==Algebraic geometry and commutative algebra==
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| In [[algebraic geometry]], a [[singular point of an algebraic variety|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 [[cusp (singularity)|cusps]]. For example, the equation <math>y^2 - x^3 = 0</math> defines a curve that has a cusp at the origin <math>x = y = 0</math>. One could define the ''x''-axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, is this case, the ''x''-axis is a "double tangent".
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| For [[affine variety|affine]] and [[projective variety|projective varieties]], the singularities are the points where the [[Jacobian matrix]] has a [[rank (linear algebra)|rank]] which is lower than at other points of the variety.
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| An equivalent definition in terms of [[commutative algebra]] may be given, which extends to [[abstract variety|abstract varieties]] and [[scheme (mathematics)|schemes]]: A point is ''singular'' if the [[Localization of a ring|local ring at this point]] is not a [[regular local ring]].
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| ==See also==
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| *[[Asymptote]]
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| *[[Catastrophe theory]]
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| *[[Defined and undefined]]
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| *[[Division by zero]]
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| *[[Hyperbolic growth]]
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| *[[Singular solution]]
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| *[[Removable singularity]]
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| *[[Vertical asymptotes]]
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| {{DEFAULTSORT:Mathematical Singularity}}
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| [[Category:Mathematical analysis]]
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