# 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

has a singularity at *z* = 0. This singularity can be removed by defining *f*(0) := 1, which is the limit of *f* as *z* tends to 0. The resulting function is holomorphic. In this case the problem was caused by *f* being given an indeterminate form. Taking a power series expansion for shows that

Formally, if is an open subset of the complex plane , a point of , and is a holomorphic function, then is called a **removable singularity** for if there exists a holomorphic function which coincides with on . We say is holomorphically extendable over if such a exists.

## Riemann's theorem

Riemann's theorem on removable singularities is as follows:

** Theorem.** Let be an open subset of the complex plane, a point of and a holomorphic function defined on the set . The following are equivalent:

- is holomorphically extendable over .
- is continuously extendable over .
- There exists a neighborhood of on which is bounded.
- .

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

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

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

We have *c*_{0} = *h*(*a*) = 0 and *c*_{1} = *h'*(*a*) = 0; therefore

Hence, where *z* ≠ *a*, we have:

However,

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:

- In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number such that . If so, is called a
**pole**of and the smallest such is the**order**of . So removable singularities are precisely the poles of order 0. A holomorphic function blows up uniformly near its other poles. - If an isolated singularity of is neither removable nor a pole, it is called an
**essential singularity**. The Great Picard Theorem shows that such an maps every punctured open neighborhood to the entire complex plane, with the possible exception of at most one point.