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
:$\text(z) = \frac$
has a singularity at . This singularity can be removed by defining $\text(0) := 1,$ which is the limit of as tends to 0. The resulting function is holomorphic. In this case the problem was caused by being given an indeterminate form. Taking a power series expansion for $\frac$ around the singular point shows that
:$\text(z) = \frac\left(\sum_^ \frac \right) = \sum_^ \frac = 1 - \frac + \frac - \frac + \cdots.$

Formally, if $U \subset \mathbb C$ is an open subset of the complex plane $\mathbb C$, $a \in U$ a point of $U$, and $f: U\setminus \ \rightarrow \mathbb C$ 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 \$. 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

:$h(z) = \begin (z - a)^2 f(z) & z \ne a ,\\ 0 & z = a . \end$

Clearly, ''h'' is holomorphic on $D \setminus \$, and there exists
:$h'(a)=\lim_\frac=\lim_(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 ''c''₀ = ''h''(''a'') = 0 and ''c''₁ = ''h''(''a'') = 0; therefore

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

Hence, where ''z'' ≠ ''a'', we have:

:$f(z) = \frac = c_2 + c_3 (z - a) + \cdots \, .$

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:

#In light of Riemann's theorem, given a non-removable singularity, one might ask whether there exists a natural number $m$ such that $\lim_(z-a)^f(z)=0$. 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.
#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.

See also

* Analytic capacity
* Removable discontinuity

External links

Removable singular point
a
Encyclopedia of Mathematics
Analytic functions
Meromorphic functions
Bernhard Riemann