# Removable singularity

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{{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](/source/parabola) with a '''removable singularity''' at {{math|1=''x'' = 2}}]]

In [complex analysis](/source/complex_analysis), a '''removable singularity''' of a [holomorphic function](/source/holomorphic_function) is a point at which the function is [undefined](/source/Undefined_(mathematics)), but it is possible to redefine the function at that point in such a way that the resulting function is [regular](/source/analytic_function) in a [neighbourhood](/source/Neighbourhood_(mathematics)) of that point.

For instance, the (unnormalized) [sinc function](/source/sinc_function), as defined by
: <math> \text{sinc}(z) = \frac{\sin z}{z} </math>
has a singularity at {{tmath|1= z = 0}}. This singularity can be removed by defining {{tmath|1= \text{sinc}(0) := 1 }}, which is the [limit](/source/Limit_of_a_function) of {{math|sinc}} as {{tmath| z }} tends to {{tmath| 0 }}. The resulting function is holomorphic. In this case the problem was caused by {{math|sinc}} being given an [indeterminate form](/source/indeterminate_form). Taking a [power series](/source/power_series) expansion for {{tmath| \textstyle \frac{\sin(z)}{z} }} 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](/source/open_subset) of the [complex plane](/source/complex_plane) {{tmath| \mathbb C }}, <math>a \in U</math> a point of {{tmath| U }}, and <math>f: U\smallsetminus \{a\} \rightarrow \mathbb C</math> is a [holomorphic function](/source/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 {{tmath| U\smallsetminus \{a\} }}. We say <math>f</math> is holomorphically extendable over <math>U</math> if such a <math>g</math> exists.

== Riemann's theorem ==

[Riemann's](/source/Bernhard_Riemann) 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 {{tmath| D \smallsetminus \{a\} }}.  The following are equivalent:
# <math>f</math> is holomorphically extendable over {{tmath| a }}. 
# <math>f</math> is continuously extendable over {{tmath| a }}. 
# There exists a [neighborhood](/source/neighborhood_(topology)) of <math>a</math> on which <math>f</math> is [bounded](/source/bounded_function).
# {{tmath|1= \lim_{z\to a}(z - a) f(z) = 0 }}.}}

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](/source/Proof_that_holomorphic_functions_are_analytic)), 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, {{tmath| h }} is holomorphic on {{tmath| D \smallsetminus \{a\} }}, 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 {{tmath| h }} is holomorphic on {{tmath| D }} and has a [Taylor series](/source/Taylor_series) about {{tmath| a }}:
: <math>h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \ldots \, .</math>

We have {{tmath|1= c_0 = h(a) = 0 }} and {{tmath|1= c_1 = h'(a) = 0}}; therefore
: <math>h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \ldots \, .</math>

Hence, where {{tmath| z \ne a }}, we have:
: <math>f(z) = \frac{h(z)}{(z - a)^2} = c_2 + c_3 (z - a) + \ldots \, .</math>

However,
: <math>g(z) = c_2 + c_3 (z - a) + \cdots \, .</math>
is holomorphic on {{tmath| D }}, thus an extension of {{tmath| f }}.

== 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 {{tmath|1= \lim_{z \rightarrow a}(z-a)^{m+1}f(z)=0 }}. If so, <math>a</math> is called a '''[pole](/source/pole_(complex_analysis))''' of <math>f</math> and the smallest such <math>m</math> is the '''order''' of {{tmath| a }}. So removable singularities are precisely the [pole](/source/pole_(complex_analysis))s of order {{tmath| 0 }}. A [meromorphic](/source/Meromorphic_function) 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](/source/essential_singularity)'''.  The [Great Picard Theorem](/source/Picard_Theorem) shows that such an <math>f</math> maps every punctured open neighborhood <math>U \smallsetminus \{a\}</math> to the entire complex plane, with the possible exception of at most one point.

== See also ==
* [Analytic capacity](/source/Analytic_capacity)
* [Removable discontinuity](/source/Removable_discontinuity)

== External links ==
* [https://www.encyclopediaofmath.org/index.php/Removable_singular_point Removable singular point] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics] {{Webarchive|url=https://archive.today/20121220135247/http://www.encyclopediaofmath.org/ |date=2012-12-20 }}

Category:Analytic functions
Category:Meromorphic functions
Category:Bernhard Riemann

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Adapted from the Wikipedia article [Removable singularity](https://en.wikipedia.org/wiki/Removable_singularity) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Removable_singularity?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.