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第1行： − {{short description|Point where a function, a curve or another mathematical object does not behave regularly}} − − 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 [[derivative|differentiability]] or [[Analyticity of holomorphic functions|analyticity]].<ref name=":0">{{Cite web|url=https://mathvault.ca/math-glossary/#singularity|title=The Definitive Glossary of Higher Mathematical Jargon — Singularity|last=|first=|date=2019-08-01|website=Math Vault|language=en-US|url-status=live|archive-url=|archive-date=|access-date=2019-12-12}}</ref><ref name=":1">{{Cite web|url=http://mathfaculty.fullerton.edu/mathews/c2003/SingularityZeroPoleMod.html|title=Singularities, Zeros, and Poles|website=mathfaculty.fullerton.edu|access-date=2019-12-12}}</ref><ref>{{Cite web|url=https://www.britannica.com/topic/singularity-complex-functions|title=Singularity {{!}} complex functions|website=Encyclopedia Britannica|language=en|access-date=2019-12-12}}</ref><ref>{{Cite web|url=https://www.thefreedictionary.com/Singularity+(mathematics)|title=Singularity (mathematics)|website=TheFreeDictionary.com|access-date=2019-12-12}}</ref> − − 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. − − − − For example, the [[function of a real variable|real function]] − − For example, the real function − − − <math> f(x)=\frac{1}{x} </math> − − = frac {1}{ x } </math > − − − − has a singularity at <math>x = 0</math>, where it seems to "explode" to <math>\pm\infty</math> and is hence not defined. The [[absolute value]] function <math>g(x) = |x|</math> also has a singularity at {{math|1=''x'' = 0}}, since it is not [[Differentiable function|differentiable]] there.<ref name=":0" /><ref>{{cite book |first=Geoffrey C. |last=Berresford |first2=Andrew M. |last2=Rockett |title=Applied Calculus |location= |publisher=Cengage Learning |year=2015 |isbn= 978-1-305-46505-3|page=151 |url=https://books.google.com/books?id=wzNBBAAAQBAJ&pg=PA151 }}</ref> − − has a singularity at <math>x = 0</math>, where it seems to "explode" to <math>\pm\infty</math> and is hence not defined. The absolute value function <math>g(x) = |x|</math> also has a singularity at , since it is not differentiable there. − − 在x=0处有一个奇点，在这里它似乎“爆炸”到±∞，因此没有定义。绝对值函数g（x）=| x |在x=0处也有奇点，因为它在那里不可微。<ref name=":0" /><ref>{{cite book |first=Geoffrey C. |last=Berresford |first2=Andrew M. |last2=Rockett |title=Applied Calculus |location= |publisher=Cengage Learning |year=2015 |isbn= 978-1-305-46505-3|page=151 |url=https://books.google.com/books?id=wzNBBAAAQBAJ&pg=PA151 }}</ref> − − The [[algebraic curve]] defined by <math>\{(x,y):y^3-x^2=0\}</math> in the <math>(x, y)</math> coordinate system has a singularity (called a [[cusp (singularity)|cusp]]) at <math>(0, 0)</math>. For singularities in [[algebraic geometry]], see [[singular point of an algebraic variety]]. For singularities in [[differential geometry]], see [[singularity theory]].+ − The algebraic curve defined by <math>\{(x,y):y^3-x^2=0\}</math> in the <math>(x, y)</math> coordinate system has a singularity (called a cusp) at <math>(0, 0)</math>. For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. − 在（x，y）坐标系中由{（x，y）：y3−x2=0}定义的代数曲线在（0,0）处有一个奇点（称为尖点）。关于代数几何中的奇点，参见代数簇中的奇点。关于微分几何中的奇点，参见奇点理论 + − ==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 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). + + − To describe the way these two types of limits are being used, 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>, then the '''left-handed limit''', <math>f(c^-)</math>, and the '''right-handed limit''', <math>f(c^+)</math>, are defined by: − To describe the way these two types of limits are being used, 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>, then the left-handed limit, <math>f(c^-)</math>, and the right-handed limit, <math>f(c^+)</math>, are defined by:+ − 为了描述这两种极限的使用方式，假设<math>f(x)</math>是实参数x的函数，对于其自变量的任意值，比如c，则左极限f（c-）和右极限f（c+）的定义如下： − :<math>f(c^-) = \lim_{x \to c}f(x)</math>, constrained by <math>x < c</math> and − + − [ math > f (c ^ -) = lim _ { x to c } f (x) </math > ，受到 < math > x </math > 和 </math > 的约束 + − :<math>f(c^+) = \lim_{x \to c}f(x)</math>, constrained by <math>x > c</math>.+ − <math>f(c^+) = \lim_{x \to c}f(x)</math>, constrained by <math>x > c</math>. − − [ math > f (c ^ +) = lim _ { x to c } f (x) </math > ，受 < math > x > c </math > 约束。 − − − − 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> . − − 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> . − − 值f（c-）是函数f（x）在值x从下接近c时趋向的值，而值f（c+）是函数f（x）在值x从上接近c时趋向的值，而不管函数在x=c点处的实际值如何 − − − There are some functions for which these limits do not exist at all. For example, the function − − There are some functions for which these limits do not exist at all. For example, the function − − 有些函数根本不存在这些限制。例如，函数 − <math>g(x) = \sin\left(\frac{1}{x}\right)</math> − + − 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 (mathematics)|undefined]]: there is no value that <math>g(x)</math> settles in on. Borrowing from complex analysis, this is sometimes called an ''[[essential singularity]]''. − 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.+ − 在x趋于<math>c = 0</math>时不趋向任何值。在这种情况下，极限不是无限的，而是没有定义的:g(x)m没有确定的值。借用复分析，这有时被称为<font color="#ff8000">本质奇点（本性奇点） essential singularity</font>。 − + − − The possible cases at a given value <math>c</math> for the argument are as follows. − − The possible cases at a given value <math>c</math> for the argument are as follows. − − 参数为给定值<math>c</math>时的可能情况如下。 − − − 连续性点是<math>c</math>的一个值，在这点f(c^-)= f（c−）= f(c^+),就像人们期望的光滑函数一样。所有的值必须是有限的。如果<math>c</math>不是连续点，则在<math>c</math>处发生不连续。 − + − + − + − ** <math>f(c)</math> has a defined value, which, however, does not match the value of the two limits. − − f（c−）≠f（c+） − 当x=c时，f（c−）没有定义；或 − − f（c−）有一个定义值，但它与两个极限的值不匹配。 − − :Type I discontinuities can be further distinguished as being one of the following subtypes: − − Type I discontinuities can be further distinguished as being one of the following subtypes: − − I型不连续性可以进一步区分为下列亚型之一: − − :* 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. − − − − :* 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). − − − * 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，都会出现可移动的不连续性。+ − + − 当f（c−）或f（c+不存在时（可能两者都不存在），就会出现“II”型不连续性。这有两个亚型，通常不单独考虑： − ** 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]]. − 无限不连续是当左极限或右极限不存在时的特例，特别是因为它是无限的，而另一个极限要么是无限的，要么是某种定义良好的有限数。换句话说，当函数的图形有一个[[垂直渐近线]]时，函数具有无限的不连续性。 − ** 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>. − “本质奇点（本性奇点）”是从复分析中借用的一个术语（见下文）。当极限f（c−）或f（c+）两者中的任意一者不存在时，情况就会如此，但不是因为它是一个“无限不连续性”。”本质奇点（本性奇点）“接近无限制，即使有效解扩展到包括<math>\pm\infty</math>。 + − 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.+ + + − 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. 第157行：
第70行： − + − 坐标奇点 − {{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 [[n-vector|{{mvar|n}}-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). − − 当在一个坐标系中出现明显的奇异性或不连续性时，就会出现<font color="#ff8000">坐标奇点 coordinate singularity</font>，可以通过选择不同的坐标系来消除。这方面的一个例子是在球面坐标系中90度纬度处的明显奇异性。在球体表面正北方移动的物体（例如，沿经度为0度的直线）将突然在极点处经历经度的瞬时变化（在本例中，从经度0跳到经度180度）。然而，这种不连续性只是显而易见的；它是所选坐标系的一个伪影，在极点处是奇异的。不同的坐标系将消除明显的不连续性（例如，用矢量表示代替经纬度表示法）。 − − − ==Complex analysis复分析== − − 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. − 在复分析中，有几类奇点。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>。+ + − + − 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 subset of the complex numbers C, with the point a being an element of U, and that f is a complex differentiable function defined on some neighborhood around a, excluding a: U \ {a}, then: − 假设 u 是复数 c 的一个开子集，点 a 是 u 的一个元素，而 f 是定义在 a 周围某个邻域上的复可微函数，除了 a: u { a } ，那么:+ − * 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>+ − 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'')<sup>''n''</sup> 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，因此奇点可移除）。 − * 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" /> − 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洛朗级数具有无穷多个负次幂。<ref name=":1" /> + + + − ===Nonisolated singularities=== − 非孤立奇点 − 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. − 簇点：孤立奇点的极限点。如果它们都是极点，尽管在每个极点上都有洛朗级数展开式，那么在极限条件下，这样的展开是不可能的。 − * '''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 [[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 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. − 分支点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分支点(如log(z)的z=0和z=∞)，这两个分支点是固定的。+ + − ==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 -->+ − 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 -->+ − [[倒数函数，显示双曲增长。[] < ! -- 一个更好的图像应该是1/(1-x)或类似的图像，显示一个正的奇点，并且随着 x 的增加而增长 -- > + + − 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]]s and PDEs ([[Partial differential equation|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 law]]s 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，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，但在奇点附近的行为通常是令人感兴趣的。在数学上，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). − − 一个例子是一个非弹性球在平面上的反弹运动。如果考虑理想化的运动，即每次弹跳动能损失的比例相同，反弹的频率就变得无限大，因为球在有限时间内静止。有限时间奇点的其他例子包括潘列夫悖论的各种形式（例如，在黑板上拖动粉笔时，粉笔会跳跃的趋势），以及在平面上旋转的硬币的进动率如何在突然停止之前加速到无限大（正如使用欧拉圆盘玩具所研究的那样）。 − − − − − 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). − − 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） − − − − − 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，代数簇的奇点是簇中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 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.+ − 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. − 对于仿射簇和射影簇，奇点是指<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 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]]. − 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，那么该点为奇点。 − − − − − − *[[Catastrophe theory]] − 突变理论 − *[[Defined and undefined]] − 定义和未定义 − *[[Degeneracy (mathematics)]] − 退化（数学 ） − *[[Division by zero]] − 除以零 − *[[Hyperbolic growth]] − 双曲线增长 − *[[Pathological (mathematics)]] − 病理学（数学） − *[[Singular solution]] − 奇异解 − *[[Removable singularity]] − 可移动奇点 − − − ==References参考== 第307行：
第152行： − {{Authority control}}+ − − − − − − Category:Mathematical analysis − − 分类: 数学分析 − − <noinclude> − − <small>This page was moved from [[wikipedia:en:Singularity (mathematics)]]. Its edit history can be viewed at [[奇点/edithistory]]</small></noinclude> − − [[Category:待整理页面]]
无编辑摘要
此词条暂由Henry翻译。已由Bai完成审校。
此词条暂由Henry翻译。已由Bai完成审校。
在数学中，<font color="#ff8000">奇点 singularity</font>一般是一个给定数学对象没有定义的点，或一个数学对象在某些特定方面不再表现良好的点，例如缺乏可微性或可分析性。<ref name=":0">{{Cite web|url=https://mathvault.ca/math-glossary/#singularity|title=The Definitive Glossary of Higher Mathematical Jargon — Singularity|last=|first=|date=2019-08-01|website=Math Vault|language=en-US|url-status=live|archive-url=|archive-date=|access-date=2019-12-12}}</ref><ref name=":1">{{Cite web|url=http://mathfaculty.fullerton.edu/mathews/c2003/SingularityZeroPoleMod.html|title=Singularities, Zeros, and Poles|website=mathfaculty.fullerton.edu|access-date=2019-12-12}}</ref><ref>{{Cite web|url=https://www.britannica.com/topic/singularity-complex-functions|title=Singularity {{!}} complex functions|website=Encyclopedia Britannica|language=en|access-date=2019-12-12}}</ref><ref>{{Cite web|url=https://www.thefreedictionary.com/Singularity+(mathematics)|title=Singularity (mathematics)|website=TheFreeDictionary.com|access-date=2019-12-12}}</ref>
在数学中，<font color="#ff8000">奇点 singularity</font>一般是一个给定数学对象没有定义的点，或一个数学对象在某些特定方面不再表现良好的点，例如缺乏可微性或可分析性。<ref name=":0">{{Cite web|url=https://mathvault.ca/math-glossary/#singularity|title=The Definitive Glossary of Higher Mathematical Jargon — Singularity|last=|first=|date=2019-08-01|website=Math Vault|language=en-US|url-status=live|archive-url=|archive-date=|access-date=2019-12-12}}</ref><ref name=":1">{{Cite web|url=http://mathfaculty.fullerton.edu/mathews/c2003/SingularityZeroPoleMod.html|title=Singularities, Zeros, and Poles|website=mathfaculty.fullerton.edu|access-date=2019-12-12}}</ref><ref>{{Cite web|url=https://www.britannica.com/topic/singularity-complex-functions|title=Singularity {{!}} complex functions|website=Encyclopedia Britannica|language=en|access-date=2019-12-12}}</ref><ref>{{Cite web|url=https://www.thefreedictionary.com/Singularity+(mathematics)|title=Singularity (mathematics)|website=TheFreeDictionary.com|access-date=2019-12-12}}</ref>
例如，实值函数
例如，实值函数
:<math> f(x)=\frac{1}{x} </math>
:<math> f(x)=\frac{1}{x} </math>
在<math>x = 0</math>处有一个奇点，在这里它似乎“爆炸”到±∞，因此没有定义。绝对值函数<math>g(x) = |x|</math>在<math>x = 0</math>处也有奇点，因为它在那里不可微。<ref name=":0" /><ref>{{cite book |first=Geoffrey C. |last=Berresford |first2=Andrew M. |last2=Rockett |title=Applied Calculus |location= |publisher=Cengage Learning |year=2015 |isbn= 978-1-305-46505-3|page=151 |url=https://books.google.com/books?id=wzNBBAAAQBAJ&pg=PA151 }}</ref>
在 <math>(x, y)</math> 坐标系中由 <math>\{(x,y):y^3-x^2=0\}</math> 定义的代数曲线在<math>(0, 0)</math>处有一个奇点（称为尖点）。关于代数几何中的奇点，参见代数簇中的奇点。关于微分几何中的奇点，参见[[奇点理论]]。
==实际分析==
在实际分析中，奇点要么是不连续的，要么是导数的不连续（有时也是高阶导数的不连续）。有四种不连续：I型，有两种亚型；II型，也可分为两种亚型（尽管通常不是）。
在实际分析中，奇点要么是不连续的，要么是导数的不连续（有时也是高阶导数的不连续）。有四种不连续：I型，有两种亚型；II型，也可分为两种亚型（尽管通常不是）。
为了描述这两种极限的使用方式，假设<math>f(x)</math>是实参数x的函数，对于其自变量的任意值，比如<math>c</math>，则左极限 <math>f(c^-)</math>和右极限<math>f(c^+)</math>的定义如下：
:<math>f(c^-) = \lim_{x \to c}f(x)</math>, constrained by <math>x < c</math>，
<math>f(c^-) = \lim_{x \to c}f(x)</math>, constrained by <math>x < c</math> and
:<math>f(c^+) = \lim_{x \to c}f(x)</math>, constrained by <math>x > c</math>。
值<math>f(c^-)</math> 是函数<math>f(x)</math>在值<math>x</math>从下接近<math>c</math>时趋向的值，而值<math>f(c^+)</math>是函数<math>f(x)</math> 在值<math>x</math>从上接近<math>c</math>时趋向的值，而不管函数在<math>x = c</math>点处的实际值如何
有些函数根本不存在这些限制。例如，函数
:<math>g(x) = \sin\left(\frac{1}{x}\right)</math>
:<math>g(x) = \sin\left(\frac{1}{x}\right)</math>
< math > g (x) = sin left (frac {1}{ x } right) </math >
在<math>x</math>趋于<math>c = 0</math>时不趋向任何值。在这种情况下，极限不是无限的，而是没有定义的：<math>g(x)</math>没有确定的值。借用复分析，这有时被称为<font color="#ff8000">本质奇点（本性奇点）essential singularity</font>。
参数为给定值<math>c</math>时的可能情况如下。
* '''连续性点'''是<math>c</math>的一个值，在这点<math>f(c^-) = f(c) = f(c^+)</math>,就像人们期望的光滑函数一样。所有的值必须是有限的。如果<math>c</math>不是连续点，则在<math>c</math>处发生不连续。
* A '''point of continuity''' is a value of <math>c</math> for which <math>f(c^-) = f(c) = f(c^+)</math>, as one expects for a smooth function. All the values must be finite. If <math>c</math> is not a point of continuity, then a discontinuity occurs at <math>c</math>.
* A '''type I''' discontinuity occurs when both <math>f(c^-)</math> and <math>f(c^+)</math> exist and are finite, but at least one of the following three conditions also applies:
* A '''type I''' discontinuity occurs when both <math>f(c^-)</math> and <math>f(c^+)</math> exist and are finite, but at least one of the following three conditions also applies:
当f（c−）和f（c+）同时存在且为有限时，即出现I型不连续，但是也至少适用以下三个条件中的一个：
当f（c−）和f（c+）同时存在且为有限时，即出现I型不连续，但是也至少适用以下三个条件中的一个：
** <math>f(c^-) \neq f(c^+)</math>;
** <math>f(c^-) \neq f(c^+)</math>
** 当<math>x = c</math>时，<math>f(x)</math> 没有定义；
** <math>f(x)</math> is not defined for the case of <math>x = c</math>; or
** 或<math>f(c)</math>有一个定义值，但它与两个极限的值不匹配。
* 当f（c−）≠f（c+）时，无论是否定义了f（c），也不管是否定义了f（c）的值，都会出现跳跃不连续。
* 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).
:I型不连续性可以进一步区分为下列亚型之一:
* 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:
:* 当<math>f(c^-) \neq f(c^+)</math>时，无论是否定义了<math>f(c)</math>，也不管是否定义了<math>f(c)</math>的值，都会出现跳跃不连续。
:* 当<math>f(c^-) = f(c^+)</math>时，无论是否定义了<math>f(c)</math>，也不管是否已定义它的值（但不匹配两个极限）时，都会出现可移动的不连续性。
* 当<math>f(c^-)</math>或<math>f(c^+)</math> 不存在时（可能两者都不存在），就会出现II型不连续性。这有两个亚型，通常不单独考虑：
** '''无限不连续'''是当左极限或右极限不存在时的特例，特别是因为它是无限的，而另一个极限要么是无限的，要么是某种定义良好的有限数。换句话说，当函数的图形有一个[[垂直渐近线]]时，函数具有无限的不连续性。
** '''本质奇点（本性奇点）'''是从复分析中借用的一个术语（见下文）。当极限 <math>f(c^-)</math>或<math>f(c^+)</math>两者中的任意一者不存在时，情况就会如此，但不是因为它是一个“无限不连续性”。”本质奇点（本性奇点）“接近无限制，即使有效解扩展到包括<math>\pm\infty</math>。
在实际分析中，奇点或不连续是函数本身的一个性质。任何可能存在于函数导数中的奇点都被认为是属于导数，而不是原函数。
在实际分析中，奇点或不连续是函数本身的一个性质。任何可能存在于函数导数中的奇点都被认为是属于导数，而不是原函数。
===Coordinate singularities===
===坐标奇点 Coordinate singularities===
当在一个坐标系中出现明显的奇异性或不连续性时，就会出现'''<font color="#ff8000">坐标奇点 coordinate singularity</font>'''，可以通过选择不同的坐标系来消除。这方面的一个例子是在球面坐标系中90度纬度处的明显奇异性。在球体表面正北方移动的物体（例如，沿经度为0度的直线）将突然在极点处经历经度的瞬时变化（在本例中，从经度0跳到经度180度）。然而，这种不连续性只是显而易见的；它是所选坐标系的一个伪影，在极点处是奇异的。不同的坐标系将消除明显的不连续性（例如，用矢量表示代替经纬度表示法）。
==复分析 Complex analysis==
===Isolated singularities孤立奇点===
在[[复分析]]中，有几类奇点。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>。
===孤立奇点 Isolated singularities===
假设 ''U'' 是复数 '''C''' 的一个开子集，点 ''a'' 是 ''U'' 的一个元素，而 ''f'' 是定义在 ''a'' 周围某个邻域上的复可微函数，除了 ''a'': ''U'' \ {''a''} ，那么:
* 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'')<sup>''n''</sup>，则点''a''为[[极点（复分析）]]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，因此奇点可移除）。
* 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洛朗级数具有无穷多个负次幂。<ref name=":1" />
===非孤立奇点 Nonisolated singularities===
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:
* '''簇点 Cluster points'''：孤立奇点的[[极限点 limit points]]。如果它们都是极点，尽管在每个极点上都有'''<font color="ff8000"[[劳伦级数 Laurent series]]</font>'''展开式，那么在极限条件下，这样的展开是不可能的。
* '''自然边界 Natural boundaries'''：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）。
===分支点 Branch points===
'''分支点 Branch point'''通常是一个多值函数的结果，如<math>\sqrt{z}</math>或<math>\log(z)</math>，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分支点(如<math>\log(z)</math>的<math>z=0</math>和<math>z=\infty</math>)，这两个分支点是固定的。
==有限时间奇点 Finite-time singularity==
[[File:Rectangular hyperbola.svg|thumb|[[倒数函数 The reciprocal function]]，显示[[双曲增长 hyperbolic growth]].]]
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，但在奇点附近的行为通常是令人感兴趣的。在数学上，最简单的有限时间奇点是<math>x^{-\alpha}</math>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：<math>x^{-1}</math> 。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 <math>(t_0-t)^{-\alpha}</math>（使用''t''表示时间，将方向反转为<math>-t</math>，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<math>t_0</math>）。
一个例子是一个非弹性球在平面上的反弹运动。如果考虑理想化的运动，即每次弹跳动能损失的比例相同，反弹的频率就变得无限大，因为球在有限时间内静止。有限时间奇点的其他例子包括[[潘列夫悖论 Painlevé paradox]]的各种形式（例如，在黑板上拖动粉笔时，粉笔会跳跃的趋势），以及在平面上旋转的硬币的进动率如何在突然停止之前加速到无限大（正如使用[[欧拉圆盘 Euler's Disk]]玩具所研究的那样）。
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 Doomsday's equation]]”（简单化模型在有限时间内产生无限多的人口）。
）
==代数几何与交换代数 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."
在代数几何中，'''代数簇的奇点'''是簇中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”。
对于[[仿射簇 affine variety]]和[[射影簇 projective variety]]，奇点是指'''<font color="#ff8000">雅可比矩阵 Jacobian matrix</font>'''的秩低于簇中其他点的秩的点。
可以给出一个关于[[交换代数 commutative algebra]]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'''奇点'''。
=参见==
*[[突变理论 Catastrophe theory]]
*[[定义和未定义 Defined and undefined]]
*[[退化（数学） Degeneracy (mathematics)]]
*[[除以零 Division by zero]]
*[[双曲线增长 Hyperbolic growth]]
*[[病理学（数学） Pathological (mathematics)]]
*[[奇异解 Singular solution]]
*[[可移动奇点 Removable singularity]]
==参考文献==
==See also=参见=
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[[Category:数学分析]]
[[Category:Mathematical analysis]]