Editing Removable singularity (section)

{{Short description|Undefined point on a holomorphic function which can be made regular}}
{{More citations needed|date=July 2021}}
[[File:Graph of x squared undefined at x equals 2.svg|thumb|right|200px|A graph of a [[parabola]] with a '''removable singularity''' at {{math|1=''x'' = 2}}]]

In [[complex analysis]], a '''removable singularity''' of a [[holomorphic function]] is a point at which the function is [[Undefined (mathematics)|undefined]], but it is possible to redefine the function at that point in such a way that the resulting function is [[analytic function|regular]] in a [[Neighbourhood (mathematics)|neighbourhood]] of that point.

For instance, the (unnormalized) [[sinc function]], as defined by
:<math> \text{sinc}(z) = \frac{\sin z}{z} </math>
has a singularity at {{math|1=''z'' = 0}}. This singularity can be removed by defining <math>\text{sinc}(0) := 1,</math> which is the [[Limit of a function|limit]] of {{math|sinc}} as {{mvar|z}} tends to 0. The resulting function is holomorphic. In this case the problem was caused by {{math|sinc}} being given an [[indeterminate form]]. Taking a [[power series]] expansion for <math display="inline">\frac{\sin(z)}{z}</math> around the singular point shows that
:<math> \text{sinc}(z) = \frac{1}{z}\left(\sum_{k=0}^{\infty} \frac{(-1)^kz^{2k+1}}{(2k+1)!} \right) = \sum_{k=0}^{\infty} \frac{(-1)^kz^{2k}}{(2k+1)!} = 1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \frac{z^6}{7!} + \cdots. </math>

Formally, if <math>U \subset \mathbb C</math> is an [[open subset]] of the [[complex plane]] <math>\mathbb C</math>, <math>a \in U</math> a point of <math>U</math>, and <math>f: U\setminus \{a\} \rightarrow \mathbb C</math> is a [[holomorphic function]], then <math>a</math> is called a '''removable singularity''' for <math>f</math> if there exists a holomorphic function <math>g: U \rightarrow \mathbb C</math> which coincides with <math>f</math> on <math>U\setminus \{a\}</math>. We say <math>f</math> is holomorphically extendable over <math>U</math> if such a <math>g</math> exists.