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

sinc (z) = \frac{\sin z}{z}

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

sinc (z) = \frac{1}{z} (\sum_{k = 0}^{\infty} \frac{(- 1)^{k} z^{2 k + 1}}{(2 k + 1)!}) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k} z^{2 k}}{(2 k + 1)!} = 1 - \frac{z^{2}}{3!} + \frac{z^{4}}{5!} - \frac{z^{6}}{7!} + \dots .

Formally, if $U \subset ℂ$ is an open subset of the complex plane ⁠ $ℂ$ ⁠, $a \in U$ a point of ⁠ $U$ ⁠, and $f : U ∖ {a} \to ℂ$ is a holomorphic function, then $a$ is called a removable singularity for $f$ if there exists a holomorphic function $g : U \to ℂ$ which coincides with $f$ on ⁠ $U ∖ {a}$ ⁠. 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:

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

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

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

h (z) = {\begin{matrix} (z - a)^{2} f (z) & z \neq a, \\ 0 & z = a . \end{matrix}

Clearly, ⁠ $h$ ⁠ is holomorphic on ⁠ $D ∖ {a}$ ⁠, and there exists

h^{'} (a) = lim_{z \to a} \frac{(z - a)^{2} f (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} + \dots .

We have ⁠ $c_{0} = h (a) = 0$ ⁠ and ⁠ $c_{1} = h^{'} (a) = 0$ ⁠; therefore

h (z) = c_{2} (z - a)^{2} + c_{3} (z - a)^{3} + \dots .

Hence, where ⁠ $z \neq a$ ⁠, we have:

f (z) = \frac{h (z)}{(z - a)^{2}} = c_{2} + c_{3} (z - a) + \dots .

However,

g (z) = c_{2} + c_{3} (z - a) + \dots .

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 $m$ such that ⁠ $lim_{z \to a} (z - a)^{m + 1} f (z) = 0$ ⁠. 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 meromorphic function blows up uniformly near its other poles.
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 $U ∖ {a}$ to the entire complex plane, with the possible exception of at most one point.

External links

Removable singular point at Encyclopedia of Mathematics Deprecated link archived 2012-12-20 at archive.today

Removable singularity

Riemann's theorem

Other kinds of singularities

See also

External links