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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 where the mathematical object ceases to be well-behaved in some particular way, such as the lack of differentiability or analyticity.

In mathematics, a singularity is in general 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 the lack of differentiability or analyticity.

==Real analysis==

==Real analysis实际分析==

 

In [[real analysis]], singularities are either [[classification of discontinuities|discontinuities]], or discontinuities of the [[derivative]] (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).

In [[real analysis]], singularities are either [[classification of discontinuities|discontinuities]], or discontinuities of the [[derivative]] (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).

* A jump discontinuity occurs when <math>f(c^-) \neq f(c^+)</math>, regardless of whether <math>f(c)</math> is defined, and regardless of its value if it is defined.

* A jump discontinuity occurs when <math>f(c^-) \neq f(c^+)</math>, regardless of whether <math>f(c)</math> is defined, and regardless of its value if it is defined.

* 当 < math > f (c ^ -) neq f (c ^ +) </math > ，不管 < math > f (c) </math > 是否定义，也不管是否定义了它的值时，就会发生跳转不连续性。

* 当f（c−）≠f（c+）时，无论是否定义了f（c），也不管定义了f（c）的值，都会出现跳跃不连续

:* A '''[[removable singularity|removable discontinuity]]''' occurs when <math>f(c^-) = f(c^+)</math>, also regardless of whether <math>f(c)</math> is defined, and regardless of its value if it is defined (but which does not match that of the two limits).

:* A '''[[removable singularity|removable discontinuity]]''' occurs when <math>f(c^-) = f(c^+)</math>, also regardless of whether <math>f(c)</math> is defined, and regardless of its value if it is defined (but which does not match that of the two limits).

* A removable discontinuity occurs when <math>f(c^-) = f(c^+)</math>, also regardless of whether <math>f(c)</math> is defined, and regardless of its value if it is defined (but which does not match that of the two limits).

* A removable discontinuity occurs when <math>f(c^-) = f(c^+)</math>, also regardless of whether <math>f(c)</math> is defined, and regardless of its value if it is defined (but which does not match that of the two limits).

* 当 < math > f (c ^ -) = f (c ^ +) </math > ，也不管 < math > f (c) </math > 是否已定义，也不管是否已定义它的值(但不匹配两个极限)时，就会出现可移动的不连续性。

* 当f（c−）≠f（c+）时，无论是否定义了f（c），也不管是否已定义它的值(但不匹配两个极限)时，就会出现可移动的不连续性。

* A '''type&nbsp;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:

* A '''type&nbsp;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:

 

** 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 of a function|graph]] has a [[vertical asymptote]].

** 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 of a function|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 <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 valid answers are extended to include <math>\pm\infty</math>.

** 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 valid answers are extended to include <math>\pm\infty</math>.

 

===Coordinate singularities===

===Coordinate singularities===

 

{{Main|Coordinate singularity}}

{{Main|Coordinate singularity}}

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).

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==

==Complex analysis复杂分析==

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

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

===Isolated singularities===

===Isolated singularities孤立奇点===

Suppose that ''U'' is an [[open set|open subset]] of the [[complex number]]s '''C''', with the point ''a'' being an element of ''U'', and that ''f'' is a [[holomorphic function|complex differentiable function]] defined on some [[Neighbourhood (mathematics)|neighborhood]] around ''a'', excluding ''a'': ''U'' \ {''a''}, then:

Suppose that ''U'' is an [[open set|open subset]] of the [[complex number]]s '''C''', with the point ''a'' being an element of ''U'', and that ''f'' is a [[holomorphic function|complex differentiable function]] defined on some [[Neighbourhood (mathematics)|neighborhood]] around ''a'', excluding ''a'': ''U'' \ {''a''}, 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'' \ {''a''}. The function ''g'' is a continuous replacement for the function ''f''.<ref>{{Cite web|url=http://mathworld.wolfram.com/Singularity.html|title=Singularity|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-12}}</ref>

* 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''.<ref>{{Cite web|url=http://mathworld.wolfram.com/Singularity.html|title=Singularity|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-12}}</ref>

 

* 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'' with ''g''(''a'') nonzero, and a [[natural number]] ''n'' such that ''f''(''z'') = ''g''(''z'') / (''z'' − ''a'')^''n'' for all ''z'' in ''U'' \ {''a''}. 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 a [[pole (complex analysis)|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]] ''n'' such that ''f''(''z'') = ''g''(''z'') / (''z'' − ''a'')^''n'' for all ''z'' in ''U'' \ {''a''}. 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 [[iff|if and only if]] the [[Laurent series]] has infinitely many powers of negative degree.<ref name=":1" />

* 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.<ref name=":1" />

 

===Nonisolated singularities===

===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:

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.

* '''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 [[analytic continuation|analytically continued]] around (or outside them if they are closed curves in the [[Riemann sphere]]).

* '''Natural boundaries''': any non-isolated set (e.g. a curve) on which functions cannot be [[analytic continuation|analytically continued]] around (or outside them if they are closed curves in the [[Riemann sphere]]).

 

===Branch points分支点===

===Branch points===

[[Branch point]]s are generally the result of a [[multi-valued function]], such as <math>\sqrt{z}</math> or <math>\log(z)</math>, 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 <math>z=0</math> and <math>z=\infty</math> for <math>\log(z)</math>) which are fixed in place.

[[Branch point]]s are generally the result of a [[multi-valued function]], such as <math>\sqrt{z}</math> or <math>\log(z)</math>, 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 <math>z=0</math> and <math>z=\infty</math> for <math>\log(z)</math>) which are fixed in place.

Branch points are generally the result of a multi-valued function, such as <math>\sqrt{z}</math> or <math>\log(z)</math>, 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 <math>z=0</math> and <math>z=\infty</math> for <math>\log(z)</math>) which are fixed in place.

Branch points are generally the result of a multi-valued function, such as <math>\sqrt{z}</math> or <math>\log(z)</math>, 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 <math>z=0</math> and <math>z=\infty</math> for <math>\log(z)</math>) which are fixed in place.

分支点通常是多值函数的结果，比如 < math > sqrt { z } </math > 或 < math > log (z) </math > ，这些分支点定义在一个特定的有限域内，因此函数可以在域内成为单值函数。切割是一条从区域中排除的直线或曲线，用以在不连续的函数值之间进行技术上的分离。当切割是真正需要的，功能将有明显不同的价值观在每一侧的分支削减。分支切口的形状是一个选择问题，即使它必须连接两个不同的分支点(比如 < math > z = 0 </math > 和 < math > z = infty </math > ，用于 < math > log (z) </math >) ，这两个分支点是固定的。

分支点通常是多值函数的结果，比如sqrt { z } 或log (z)，这些分支点定义在一个特定的有限域内，因此函数可以在域内成为单值函数。切割是一条从区域中排除的直线或曲线，用以在不连续的函数值之间进行技术上的分离。当真正需要切割时，函数将在分支切割的每一侧具有明显不同的值。。分支切口的形状是一个选择问题，即使它必须连接两个不同的分支点(比如 z = 0 和z = infty ，用于log (z)) ，这两个分支点是固定的

==Finite-time singularity==

==Finite-time singularity有限时间奇点==

[[File: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 -->

[[File: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 -->

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 kinematics and PDEs (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 <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 that time increases to infinity, and shifting the singularity forward from 0 to a fixed time <math>t_0</math>).

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 kinematics and PDEs (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 <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 that time increases to infinity, and shifting the singularity forward from 0 to a fixed time <math>t_0</math>).

当输入变量为时间，输出变量在有限时间内向无穷大方向增长时，出现有限时间奇异性。这些在运动学和偏微分方程(偏微分方程)中都很重要——物理上不存在无穷大，但奇点附近的行为通常是有趣的。数学上，最简单的有限时间奇点是形式为 < math > x ^ {-alpha }的各种指数的幂律，其中最简单的是双曲增长，其指数是(负)1: < math > x ^ {-1}。更准确地说，为了在正时间得到一个奇点，随着时间的推移(所以输出增加到无穷大) ，一个代替使用 < math > (t _ 0-t) ^ {-alpha } </math > (使用 t 表示时间，倒转方向到 < math >-t </math > ，所以时间增加到无穷大，并将奇点从0向前移动到一个固定的时间 < t _ 0 </math >)。

当一个输入变量为时间时，出现有限时间奇点，而输出变量在有限时间向无穷大方向增加。这些在运动学和偏微分方程（偏微分方程）中很重要——无穷大在物理上并不存在，但在奇点附近的行为通常是令人感兴趣的。从数学上讲，最简单的有限时间奇点是x-α形式的各种指数的幂律，其中最简单的是双曲增长，其中指数为（负）1:x−1。更准确地说，为了随着时间的推移在正时间处获得奇点（因此输出增长到无穷大），可以使用（t0−t）−α（使用t表示时间，将方向反转为−t，以便时间增加到无穷大，并将奇点从0向前移动到固定时间t0）。

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 various forms of the Painlevé paradox (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).

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 various forms of the Painlevé paradox (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).

Hypothetical examples include Heinz von Foerster's facetious "Doomsday's equation" (simplistic models yield infinite human population in finite time).

Hypothetical examples include Heinz von Foerster's facetious "Doomsday's equation" (simplistic models yield infinite human population in finite time).

假设的例子包括 Heinz von Foerster 滑稽的“末日方程”(简单化的模型在有限的时间里产生无限的人口)。

假设的例子包括海因茨·冯·福斯特（Heinz von Foerster）滑稽的“世界末日方程”（简单化模型在有限时间内产生无限多的人口）

==Algebraic geometry and commutative algebra==

==Algebraic geometry and commutative algebra代数几何与交换代数==

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|1= ''y''{{sup|2}} − ''x''{{sup|3}} = 0 }} defines a curve that has a cusp at the origin {{math|1= ''x'' = ''y'' = 0 }}. One could define the {{math|''x''}}-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 {{math|''x''}}-axis is a "double tangent."

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|1= ''y''{{sup|2}} − ''x''{{sup|3}} = 0 }} defines a curve that has a cusp at the origin {{math|1= ''x'' = ''y'' = 0 }}. One could define the {{math|''x''}}-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 {{math|''x''}}-axis is a "double tangent."

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  − x = 0 }} 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."

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  − x = 0 }} 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."

在代数几何中，代数簇的奇点是各种各样的切线空间可能没有规则定义的一点。奇点最简单的例子就是它们自己交叉的曲线。但是还有其他类型的奇点，比如尖点。例如，方程 -x = 0}}定义了一条在原点有尖点的曲线。可以将-轴定义为这一点的切线，但这个定义不能与其他点的定义相同。实际上，在这种情况下,-轴是一个“双切线”

在代数几何中，代数簇的奇点是各种各样的切线空间可能没有规则定义的一点。奇点最简单的例子就是它们自己交叉的曲线。但是还有其他类型的奇点，比如尖点。例如，方程 -x = 0定义了一条在原点有尖点的曲线。可以将-轴定义为这一点的切线，但这个定义不能与其他点的定义相同。实际上，在这种情况下,-轴是一个“双切线”

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.

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.

==See also==

==See also=参见=

*[[Catastrophe theory]]

*[[Catastrophe theory]]

 

*[[Defined and undefined]]

*[[Defined and undefined]]

 

*[[Degeneracy (mathematics)]]  

*[[Degeneracy (mathematics)]]  

 

*[[Division by zero]]

*[[Division by zero]]

 

*[[Hyperbolic growth]]

*[[Hyperbolic growth]]

 

*[[Pathological (mathematics)]]

*[[Pathological (mathematics)]]

 

*[[Singular solution]]

*[[Singular solution]]

 

*[[Removable singularity]]

*[[Removable singularity]]

 

==References参考==

==References==

{{Reflist}}

{{Reflist}}