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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).

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.

在复分析中，有几类<font color="#ff8000">奇点</font>。其中包括孤立奇点、非孤立奇点和分支点。

在复分析中，有几类<font color="#ff8000">奇点</font>。其中包括<font color="#ff8000">孤立奇点  isolated singularities</font>、<font color="#ff8000">非孤立奇点  nonisolated singularities</font> 和<font color="#ff8000">分支点 branch points</font>。

如果存在一个定义在所有U上的全纯函数g，使得对于U \ {a}中的所有z, f(z) = g(z)，那么点a是f的一个可去奇点。函数g是函数f的连续替换。<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>

如果存在一个定义在所有U上的全纯函数g，使得对于U \ {a}中的所有z, f(z) = g(z)，那么点a是f的一个可去奇点。函数g是函数f的连续替换。<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).

如果存在定义在“U”上的全纯函数“g”，且“g”（“a”）非零，且存在一个自然数“n”，使得对所有“z”属于“U”\{“a”},“f”（“z”）=“g”（“z”）/ (“z” – “a”)n，则点“a”为[[极点（复分析）|极]]或“f”的非本质奇点。最小的这个数“n”称为“极序”。非本质奇点处的导数本身也有一个非本质奇点，当“n”增加1时（除非“n”为0，因此<font color="#ff8000">奇点</font>可移除）。  

如果存在定义在“U”上的全纯函数“g”，且“g”（“a”）非零，且存在一个自然数“n”，使得对所有“z”属于“U”\{“a”},“f”（“z”）=“g”（“z”）/ (“z” – “a”)n，则点“a”为[[极点（复分析）|极]]或“f”的<font color="#ff8000">非本质奇点</font>。最小的这个数“n”称为“极序”。非本质奇点处的导数本身也有一个非本质奇点，当“n”增加1时（除非“n”为0，因此<font color="#ff8000">奇点</font>可移除）。  

* 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" />

如果点“a”既不是可去奇点，也不是极点，则它是“f”的 非本质奇点。点“a”是非本质奇点当且仅当[[Laurent级数]]具有无穷多个负次幂。

如果点“a”既不是可去奇点，也不是极点，则它是“f”的<font color="#ff8000">非本质奇点</font>。点“a”是<font color="#ff8000">非本质奇点</font>当且仅当洛朗级数具有无穷多个负次幂。

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:

除孤立奇点外，一个变量的复变函数还可能表现出其他奇异行为。这些称为非孤立奇点，其中有两种类型:

除<font color="#ff8000">孤立奇点</font>外，一个变量的复变函数还可能表现出其他奇异行为。这些称为<font color="#ff8000">非孤立奇点</font>，其中有两种类型:

* '''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]]).

自然边界：函数不能在其上[[解析延拓|解析连续]]围绕的任何非孤立集（如曲线）（如果它们是[[黎曼球面]]中的闭合曲线，则在其外部）。

自然边界：函数不能 [[解析延拓|解析连续]]在其周围（或在其外部，如果它们是黎曼球面上的闭合曲线）的任何非孤立集合（如曲线）。

===Branch points分支点===

===Branch points分支点===

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

<font color="#ff8000">分支点</font>通常是一个多值函数的结果，如z√或log(z)，它们被定义在一个特定的限定域中，这样函数就可以在该域中成为单值函数。切线是被排除在域之外的一条线或曲线，用于在不连续的函数值之间引入技术分离。当真正需要切割时，该函数在分支切割的每一边都有明显不同的值。分枝切割的形状是一个选择的问题，即使它必须连接两个不同的<font color="#ff8000">分支点</font>(如log(z)的z=0和z=∞)，这两个<font color="#ff8000">分支点</font>是固定的。

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）。

当一个输入变量为时间时，而一个输出变量在有限时间趋于无穷大时，就会出现<font color=“#ff8000”>有限时间奇点 finite-time singularity </font>，。这些在运动学和偏微分方程中很重要——无穷大在物理上并不存在，但在<font color=“#ff8000”>奇点</font>附近的行为通常是令人感兴趣的。在数学上，最简单的<font color=“#ff8000”>有限时间奇点</font>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<font color=“#ff8000”>奇点</font>（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<font color=“#ff8000”>奇点</font>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).

 

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<font color=“#ff8000”>奇点</font>是簇中切线空间可能没有规则定义的一点。<font color=“#ff8000”>奇点</font>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<font color=“#ff8000”>奇点</font>，比如尖点。例如，方程 -x = 0定义了一条在原点有一个尖点的曲线。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.

对于仿射变种和射影变种，奇异点是指<font color="#ff8000"> 雅可比矩阵Jacobian matrix</font>的秩低于其它变种点的秩的点。

对于仿射簇和射影簇，<font color=“#ff8000”>奇点</font>是指<font color="#ff8000"> 雅可比矩阵Jacobian matrix</font>的秩低于簇中其他点的秩的点。

An equivalent definition in terms of commutative algebra 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.

An equivalent definition in terms of commutative algebra 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.

可以给出一个关于交换代数的等价定义，它扩展到抽象的簇和方案: 如果局部环在这一点上不是一个正则局部环，那么一个点就是奇异的。

可以给出一个关于交换代数的等价定义，它扩展到抽象的簇和[[方案]]: 如果局部环在这一点上不是一个正则局部环，那么该点为<font color=“#ff8000”>奇点</font>。