# Removable singularity

Short description: Undefined point on a holomorphic function which can be made regular
A graph of a parabola with a removable singularity at 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

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

has a singularity at z = 0. This singularity can be removed by defining $\displaystyle{ \text{sinc}(0) := 1, }$ 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 $\displaystyle{ \frac{\sin(z)}{z} }$ around the singular point shows that

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

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

## Riemann's theorem

Riemann's theorem on removable singularities is as follows:

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

1. $\displaystyle{ f }$ is holomorphically extendable over $\displaystyle{ a }$.
2. $\displaystyle{ f }$ is continuously extendable over $\displaystyle{ a }$.
3. There exists a neighborhood of $\displaystyle{ a }$ on which $\displaystyle{ f }$ is bounded.
4. $\displaystyle{ \lim_{z\to a}(z - a) f(z) = 0 }$.

The implications 1 ⇒ 2 ⇒ 3 ⇒ 4 are trivial. To prove 4 ⇒ 1, we first recall that the holomorphy of a function at $\displaystyle{ a }$ is equivalent to it being analytic at $\displaystyle{ a }$ (proof), i.e. having a power series representation. Define

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

Clearly, h is holomorphic on $\displaystyle{ D \setminus \{a\} }$, and there exists

$\displaystyle{ h'(a)=\lim_{z\to a}\frac{(z - a)^2f(z)-0}{z-a}=\lim_{z\to a}(z - a) f(z)=0 }$

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

$\displaystyle{ h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, . }$

We have c0 = h(a) = 0 and c1 = h'(a) = 0; therefore

$\displaystyle{ h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \cdots \, . }$

Hence, where $\displaystyle{ z \ne a }$, we have:

$\displaystyle{ f(z) = \frac{h(z)}{(z - a)^2} = c_2 + c_3 (z - a) + \cdots \, . }$

However,

$\displaystyle{ g(z) = c_2 + c_3 (z - a) + \cdots \, . }$

is holomorphic on D, thus an extension of $\displaystyle{ f }$.

## 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:

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