In mathematics a singularity is a point at which a mathematical object (e.g., a function) is not defined or behaves `badly’ in some way. Singularities can be isolated (e.g., removable singularities, poles and essential singularities) or nonisolated (e.g., branch cuts). For teaching purposes, I want to delve into some of the mathematical aspects of isolated singularities in this note using simple examples involving the complex sine function. I will not consider nonisolated singularities in detail. These are briefly discussed with some examples in this Wikipedia page. I will also briefly look at how singularities arise in the context of black hole physics in a short final section.
Definition: A function has an isolated singularity at the point
if
is analytic on a punctured open disc
, where
, but not at
itself.
Note that a function is analytic at a point
if it is differentiable on a region containing
. Strangely, a function can have a derivative at a point without being analytic there. For example, the function
has a derivative at
but at no other point, as can easily be verified using the Cauchy-Riemann equations. Therefore this function is not analytic at
. Also note with regard to the definition of an isolated singularity that the function MUST be analytic on the `whole’ of the punctured open disc for the singularity to be defined. For example, despite appearances, the function
does not have a singularity at because it is impossible to define a punctured open disc centred at
on which
is analytic (the function
is discontinuous everywhere on the negative real axis, so
fails to be analytic there).
I find it appealing that all three types of isolated singularity (removable, poles and essential singularities) can be illustrated by using members of the following family of functions:
where . For example, if
we get
which has a removable singularity at . If
we get
which has a pole of order at
. Finally, if
we get
which has an essential singularity at . In each of these three cases, the function is not analytic at
but is analytic on a punctured open disc with centre
, e.g.,
or indeed
(which can be thought of as a punctured disc with infinite radius). In what follows I will use these three examples to delve into structural definitions of the three types of singularity. I will then explore their classification using Laurent series expansions.
Structural definitions of isolated singularities
Removable singularities
Suppose a function is analytic on the punctured open disc
and has a singularity at . The function
has a removable singularity at
if there is a function
which is analytic at
such that
for
We can see that extends the analyticity of
to include
, so we say that
is an analytic extension of
to the circle
With removable singularities we always have that exists since
(this will not be true for the other types of singularity) and the name of this singularity comes from the fact that we can effectively `remove’ the singularity by defining .
To apply this to the function
we first observe that the Maclaurin series expansion of is
for
Therefore we can write
for
If we then set
for
we see that extends the analyticity of
to include
. We also see that
Therefore has a removable singularity at
.
Poles of order k, k > 0
Suppose a function is analytic on the punctured open disc
and has a singularity at . The function
has a pole of order
at
if there is a function
, analytic at
with
, such that
for
With poles of order we always have that
as
(which distinguishes them from removable singularities)
and
exists and is nonzero (since ).
To apply this to the function
we first observe that
for
where is the function
for
Since , we see that
behaves like
near
and
as
so the singularity at is not removable. We also see that
Therefore the function has a pole of order
at
.
Essential singularities
Suppose a function is analytic on the punctured open disc
and has a singularity at . The function
has an essential singularity at
if the singularity is neither removable nor a pole. Such a singularity cannot be removed in any way, including by mutiplying by any
, hence the name.
With essential singularities we have that
does not exist, and does not tend to infinity as
.
To apply this to the function
we observe that if we restrict the function to the real axis and consider a sequence of points
then we have that whereas
Therefore
does not exist, so the singularity is not removable, but it is also the case that
so the singularity is not a pole. Since it is neither a removable singularity nor a pole, it must be an essential singularity.
Classification of isolated singularities using Laurent series
By Laurent’s Theorem, a function which is analytic on an open annulus
(shown in the diagram) can be represented as an extended power series of the form
for , which converges at all points in the annulus. It is an `extended’ power series because it involves negative powers of
. (The part of the power series involving negative powers is often referred to as the singular part. The part involving non-negative powers is referred to as the analytic part). This extended power series representation is the Laurent series about
for the function
on the annulus
. Laurent series are also often used in the case when
is a punctured open disc, in which case we refer to the series as the Laurent series about
for the function
.
The Laurent series representation of a function on an annulus is unique. We can often use simple procedures, such as finding ordinary Maclaurin or Taylor series expansions, to obtain an extended power series and we can feel safe in the knowledge that the power series thus obtained must be the Laurent series.
Laurent series expansions can be used to classify singularities by virtue of the following result: If a function has a singularity at
and if its Laurent series expansion about
is
then
(a) has a removable singularity at
iff
for all
;
(b) has a pole of order
at
iff
for all
and
;
(c) has an essential singularity at
iff
for infinitely many
.
To apply this to our three examples, observe that the function
has a singularity at and its Laurent series expansion about
is
for . This has no non-zero coefficients in its singular part (i.e., it only has an analytic part) so the singularity is a removable one.
The function
has a singularity at and its Laurent series expansion about
is
for . This has
for all
and
, so the singularity in this case is a pole of order
.
Finally, the function
has a singularity at and its Laurent series expansion about
is
for . This has
for infinitely many
so the singularity here is an essential singularity.
Singularities in Schwarzschild black holes
One often hears about singularities in the context of black hole physics and I wanted to quickly look at singularities in the particular case of non-rotating black holes. A detailed investigation of the various singularities that appear in exact solutions of Einstein’s field equations was conducted in the 1960s and 1970s by Penrose, Hawking, Geroch and others. See, e.g., this paper by Penrose and Hawking. There is now a vast literature on this topic. The following discussion is just my own quick look at how the ideas might arise.
The spacetime of a non-rotating spherical black hole is usually analysed using the Schwarzschild solution of the Einstein field equations for an isolated spherical mass . In spherical coordinates this is the metric
where
and
is the mass of the spherically symmetric static object exterior to which the Schwarzschild metric applies. If we consider only radial motion (i.e., world lines for which
) the Schwarzschild metric simplifies to
We can see that the term in the metric becomes infinite at
so there is apparently a singularity here. However, this singularity is `removable’ by re-expressing the metric in a new set of coordinates,
and
, known as the Eddington-Finkelstein coordinates. The transformed metric has the form
which does not behave badly at . In general relativity, this type of removable singularity is known as a coordinate singularity. Another example is the apparent singularity at the
latitude in spherical coordinates, which disappears when a different coordinate system is used.
Since the term in the Schwarzschild metric becomes infinite at
, it appears that we also have a singularity at this point. This is not a removable singularity and can in fact be recognised in terms of the earlier discussion above as a pole of order 1 (also called a simple pole).