Editing Singularity (mathematics) (section)

==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 <math>f</math> is a function that is [[holomorphic function|complex differentiable]] in the [[set complement|complement]] of a point <math>a</math> in an [[open set|open subset]] <math>U</math> of the [[complex number]]s <math>\mathbb{C}.</math> Then:
* The point <math>a</math> is a [[removable singularity]] of <math>f</math> if there exists a [[holomorphic function]] <math>g</math> defined on all of <math>U</math> such that <math>f(z) = g(z)</math> for all <math>z</math> in <math>U \smallsetminus \{ a \}.</math> The function <math>g</math> is a continuous replacement for the function <math>f.</math><ref name=mathworld>{{cite web |last=Weisstein |first=Eric W. |title=Singularity |website=mathworld.wolfram.com |lang=en |url=http://mathworld.wolfram.com/Singularity.html |access-date=2019-12-12}}</ref>
* The point <math>a</math> is a [[pole (complex analysis)|pole]] or non-essential singularity of <math>f</math> if there exists a holomorphic function <math>g</math> defined on <math>U</math> with <math>g(a)</math> nonzero, and a [[natural number]] <math>n</math> such that <math>f(z) = \frac{ g(z) }{(z-a)^n}</math> for all <math>z</math> in <math>U \smallsetminus \{ a \}.</math> The least such number <math>n</math> is called the ''order of the pole''. The derivative at a non-essential singularity itself has a non-essential singularity, with <math>n</math> increased by {{math|1}} (except if <math>n</math> is {{math|0}} so that the singularity is removable).
* The point <math>a</math> is an [[essential singularity]] of <math>f</math> if it is neither a removable singularity nor a pole. The point <math>a</math> is an essential singularity [[iff|if and only if]] the [[Laurent series]] has infinitely many powers of negative degree.<ref name=":1"/>

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

===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.