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Removable singularity

Undefined point on a holomorphic function which can be made regular

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated December 14, 2025
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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

{\text{sinc}}(z)={\frac {\sin z}{z}}

has a singularity at ⁠z=0⁠. This singularity can be removed by defining ⁠{\text{sinc}}(0):=1⁠, which is the limit of sinc as ⁠z⁠ tends to ⁠0⁠. The resulting function is holomorphic. In this case the problem was caused by sinc being given an indeterminate form. Taking a power series expansion for ⁠\textstyle {\frac {\sin(z)}{z}}⁠ around the singular point shows that

{\text{sinc}}(z)={\frac {1}{z}}\left(\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{(2k+1)!}}\right)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k+1)!}}=1-{\frac {z^{2}}{3!}}+{\frac {z^{4}}{5!}}-{\frac {z^{6}}{7!}}+\cdots .

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

01Riemann's theorem

Riemann's theorem on removable singularities is as follows:

Theorem, Let D\subset \mathbb {C} be an open subset of the complex plane, a\in D a point of D and f a holomorphic function defined on the set ⁠D\smallsetminus \{a\}⁠. The following are equivalent:

  1. f is holomorphically extendable over ⁠a⁠.
  2. f is continuously extendable over ⁠a⁠.
  3. There exists a neighborhood of a on which f is bounded.
  4. ⁠\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 a is equivalent to it being analytic at a (proof), i.e. having a power series representation. Define

h(z)={\begin{cases}(z-a)^{2}f(z)&z\neq a,\\0&z=a.\end{cases}}

Clearly, ⁠h⁠ is holomorphic on ⁠D\smallsetminus \{a\}⁠, and there exists

h'(a)=\lim _{z\to a}{\frac {(z-a)^{2}f(z)-0}{z-a}}=\lim _{z\to a}(z-a)f(z)=0

by 4, hence ⁠h⁠ is holomorphic on ⁠D⁠ and has a Taylor series about ⁠a⁠:

h(z)=c_{0}+c_{1}(z-a)+c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\ldots \,.

We have ⁠c_{0}=h(a)=0⁠ and ⁠c_{1}=h'(a)=0⁠; therefore

h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\ldots \,.

Hence, where ⁠z\neq a⁠, we have:

f(z)={\frac {h(z)}{(z-a)^{2}}}=c_{2}+c_{3}(z-a)+\ldots \,.

However,

g(z)=c_{2}+c_{3}(z-a)+\cdots \,.

is holomorphic on ⁠D⁠, thus an extension of ⁠f⁠.

02Other 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:

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

This article is adapted from the Wikipedia article “Removable singularity”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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