{{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, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a 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 {{tmath|1= z = 0}}. This singularity can be removed by defining {{tmath|1= \text{sinc}(0) := 1 }}, which is the limit 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. Taking a 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 of the 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, 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 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 of <math>a</math> on which <math>f</math> is bounded. # {{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), 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 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''' of <math>f</math> and the smallest such <math>m</math> is the '''order''' of {{tmath| a }}. So removable singularities are precisely the poles of order {{tmath| 0 }}. A meromorphic 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 Great 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 * 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