Editing Removable singularity

{{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.

== Riemann's theorem ==

[[Bernhard Riemann|Riemann's]] theorem on removable singularities is as follows:

{{math theorem| Let <math>D \subset \mathbb C</math> be an open subset of the complex plane, <math>a \in D</math> a point of <math>D</math> and <math>f</math> a holomorphic function defined on the set <math>D \setminus \{a\}</math>.  The following are equivalent:

# <math>f</math> is holomorphically extendable over <math>a</math>. 
# <math>f</math> is continuously extendable over <math>a</math>. 
# There exists a [[neighborhood (topology)|neighborhood]] of <math>a</math> on which <math>f</math> is [[bounded function|bounded]].
# <math>\lim_{z\to a}(z - a) f(z) = 0</math>.}}

The implications 1 ⇒ 2  ⇒ 3  ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at <math>a</math> is equivalent to it being analytic at <math>a</math> ([[Proof that holomorphic functions are analytic|proof]]), i.e. having a power series representation. Define

:<math>
  h(z) = \begin{cases}
    (z - a)^2 f(z) &  z \ne a ,\\
    0              &  z = a .
  \end{cases}
</math>

Clearly, ''h'' is holomorphic on <math> D \setminus \{a\}</math>, and there exists
:<math>h'(a)=\lim_{z\to a}\frac{(z - a)^2f(z)-0}{z-a}=\lim_{z\to a}(z - a) f(z)=0</math>
by 4, hence ''h'' is holomorphic on ''D'' and has a [[Taylor series]] about ''a'':

:<math>h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, .</math>

We have ''c''<sub>0</sub> = ''h''(''a'') = 0 and ''c''<sub>1</sub> = ''h{{'}}''(''a'') = 0; therefore

:<math>h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, .</math>

Hence, where <math>z \ne a</math>, we have:

:<math>f(z) = \frac{h(z)}{(z - a)^2} = c_2 + c_3 (z - a) + \cdots \, .</math>

However,

:<math>g(z) = c_2 + c_3 (z - a) + \cdots \, .</math>

is holomorphic on ''D'', thus an extension of <math> f </math>.

== Other kinds of singularities ==

Unlike functions of a real variable, holomorphic functions are sufficiently rigid that their isolated singularities can be completely classified. A holomorphic function's singularity is either not really a singularity at all, i.e. a removable singularity, or one of the following two types:

#In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number <math>m</math> such that <math>\lim_{z \rightarrow a}(z-a)^{m+1}f(z)=0</math>. If so, <math>a</math> is called a '''[[pole (complex analysis)|pole]]''' of <math>f</math> and the smallest such <math>m</math> is the '''order''' of <math>a</math>. So removable singularities are precisely the [[pole (complex analysis)|pole]]s of order 0. A holomorphic function blows up uniformly near its other poles.
#If an isolated singularity <math>a</math> of <math>f</math> is neither removable nor a pole, it is called an '''[[essential singularity]]'''.  The [[Picard Theorem|Great Picard Theorem]] shows that such an <math>f</math> maps every punctured open neighborhood <math>U \setminus \{a\}</math> to the entire complex plane, with the possible exception of at most one point.

==See also==
* [[Analytic capacity]]
* [[Removable discontinuity]]

== External links ==
*[https://www.encyclopediaofmath.org/index.php/Removable_singular_point Removable singular point] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
[[Category:Analytic functions]]
[[Category:Meromorphic functions]]
[[Category:Bernhard Riemann]]