Removable singularity



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 $x = 2$. This singularity can be removed by defining $$\text{sinc}(0) := 1,$$ which is the limit of $z = 0$ 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 $\frac{\sin(z)}{z}$ around the singular point shows that
 * $$ \text{sinc}(z) = \frac{1}{z}\left(\sum_{k=0}^{\infty} \frac{(-1)^kz^{2k+1}}{(2k+1)!} \right) = \sum_{k=0}^{\infty} \frac{(-1)^kz^{2k}}{(2k+1)!} = 1 - \frac{z^2}{3!} + \frac{z^4}{5!} - \frac{z^6}{7!} + \cdots. $$

Formally, if $$U \subset \mathbb C$$ is an open subset of the complex plane $$\mathbb C$$, $$a \in U$$ a point of $$U$$, and $$f: U\setminus \{a\} \rightarrow \mathbb C$$ 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 $$U\setminus \{a\}$$. We say $$f$$ is holomorphically extendable over $$U$$ if such a $$g$$ exists.

Riemann's theorem
Riemann's theorem on removable singularities is as follows:

$$

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 \ne a ,\\ 0             &  z = a. \end{cases} $$

Clearly, h is holomorphic on $$ D \setminus \{a\}$$, and there exists
 * $$h'(a)=\lim_{z\to a}\frac{(z - a)^2f(z)-0}{z-a}=\lim_{z\to a}(z - a) f(z)=0$$

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


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

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


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

Hence, where $$z \ne a$$, we have:


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

However,


 * $$g(z) = c_2 + c_3 (z - a) + \cdots \, .$$

is holomorphic on D, thus an extension of $$ f $$.

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 $$m$$ 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 holomorphic 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 $$U \setminus \{a\}$$ to the entire complex plane, with the possible exception of at most one point.