Definitions

Removable singularity

In complex analysis, a removable singularity of a holomorphic function is a point at which the function is ostensibly undefined, but, upon closer examination, the domain of the function can be enlarged to include the singularity (in such a way that the function remains holomorphic).

For instance, the function

$f\left(z\right) = frac\left\{sin z\right\}\left\{z\right\}$

for z ≠ 0 has a singularity at z = 0. This singularity can be removed by defining f(0) = 1. The resulting function is a continuous, in fact holomorphic, function.

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

Riemann's theorem

Riemann's theorem on removable singularities states when a singularity is removable:

Theorem. The following are equivalent:

i) f is holomorphically extendable over a.

ii) f is continuously extendable over a.

iii) There exists a neighborhood of a on which f is bounded.

iv) limza(z - a) f(z) = 0.

The implications i) ⇒ ii) ⇒ iii) ⇒ iv) are trivial. To prove iv) ⇒ i), we first recall that the holomorphy of a function at a is equivalent to it being analytic at a, 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}

Then

$h\left(z\right) - h\left(a\right) = \left(z - a\right)\left(z - a\right)f\left(z\right), ,$

where, by assumption, (z - a)f(z) can be viewed as a continuous function on D. In other words, h is holomorphic on D and has a Taylor series about a:

$h\left(z\right) = a_2 \left(z - a\right)^2 + a_3 \left(z - a\right)^3 + cdots .$

Therefore

$g\left(z\right) = frac\left\{h\left(z\right)\right\}\left\{\left(z-a\right)^2\right\}$

is a holomorphic extension of f over a, which proves the claim.

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 limza(z - a )m+1f(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 poles.
2. If an isolated singularity a of f is neither removable nor a pole, it is called an essential singularity. It can be shown that such an f maps every punctured open neighborhood U - {a} to the entire complex plane, with the possible exception of at most one point.