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 has a singularity at z = 0.

This singularity can be removed by defining

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

around the singular point shows that Formally, if

is an open subset of the complex plane

is called a removable singularity for

if there exists a holomorphic function

be an open subset of the complex plane,

a holomorphic function defined on the set

The following are equivalent: The implications 1 ⇒ 2 ⇒ 3 ⇒ 4 are trivial.

To prove 4 ⇒ 1, we first recall that the holomorphy of a function at

(proof), i.e. having a power series representation.

, and there exists by 4, hence h is holomorphic on D and has a Taylor series about a: We have c0 = h(a) = 0 and c1 = h'(a) = 0; therefore Hence, where

Unlike functions of a real variable, holomorphic functions are sufficiently rigid that their isolated singularities can be completely classified.