Editing Singularity (mathematics) (section)

===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"/>