3Laurent series and singularities
IB Complex Methods
3.3 Classification of singularities
The previous section was rather boring — you’ve probably seen all of that before.
It is just there as a buildup for our study of singularities. These are in some
sense the “opposites” of zeros.
Definition
(Isolated singularity)
.
Suppose that
f
has a singularity at
z
0
=
z
.
If there is a neighbourhood of
z
0
within which
f
is analytic, except at
z
0
itself,
then
f
has an isolated singularity at
z
0
. If there is no such neighbourhood, then
f has an essential (non-isolated) singularity at z
0
.
Example. cosech z
has isolated singularities at
z
=
nπi, n ∈ Z
, since
sinh
has
zeroes at these points.
Example. cosech
1
z
has isolated singularities at
z
=
1
nπi
, with
n 6
= 0, and an
essential non-isolated singularity at
z
= 0 (since there are other arbitrarily close
singularities).
Example. cosech z
also has an essential non-isolated singularity at
z
=
∞
, since
cosech
1
z
has an essential non-isolated singularity at z = 0.
Example. log z
has a non-isolated singularity at
z
= 0, because it is not analytic
at any point on the branch cut. This is normally referred to as a branch point
singularity.
If
f
has an isolated singularity, at
z
0
, we can find an annulus 0
< |z −z
0
| < r
within which
f
is analytic, and it therefore has a Laurent series. This gives us a
way to classify singularities:
(i) Check for a branch point singularity.
(ii) Check for an essential (non-isolated) singularity.
(iii)
Otherwise, consider the coefficients of the Laurent series
P
∞
n=−∞
a
n
(
z −
z
0
)
n
:
(a) If a
n
= 0 for all n < 0, then f has a removable singularity at z
0
.
(b)
If there is a
N >
0 such that
a
n
= 0 for all
n < −N
but
a
−N
6
= 0,
then
f
has a pole of order
N
at
z
0
(for
N
= 1
,
2
, ···
, this is also called
a simple pole, double pole etc.).
(c)
If there does not exist such an
N
, then
f
has an essential isolated
singularity.
A removable singularity (one with Laurent series
a
0
+
a
1
(
z −z
0
)+
···
) is so called
because we can remove the singularity by redefining
f
(
z
0
) =
a
0
=
lim
z→z
0
f
(
z
);
then f will become analytic at z
0
.
Let’s look at some examples. In fact, we have 10 examples here.
Example.
(i)
1
z−i
has a simple pole at
z
=
i
. This is since its Laurent series is, err,
1
z−i
.
(ii)
cos z
z
has a singularity at the origin. This has Laurent series
cos z
z
= z
−1
−
1
2
z +
1
24
z
3
− ··· ,
and hence it has a simple pole.
(iii)
Consider
z
2
(z−1)
3
(z−i)
2
. This has a double pole at
z
=
i
and a triple pole at
z
= 1. To show formally that, for instance, there is a double pole at
z
=
i
,
notice first that
z
2
(z−1)
3
is analytic at z = i. So it has a Taylor series, say,
b
0
+ b
1
(z − i) + b
2
(z − i)
2
+ ···
for some
b
n
. Moreover, since
z
2
(z−1)
3
is non-zero at
z
=
i
, we have
b
0
6
= 0.
Hence
z
2
(z − 1)
3
(z − i)
2
=
b
0
(z − i)
2
+
b
1
z − i
+ b2 + ··· .
So this has a double pole at z = i.
(iv)
If
g
(
z
) has zero of order
N
at
z
=
z
0
, then
1
g(z)
has a pole of order
N
there,
and vice versa. Hence cot z has a simple pole at the origin, because tan z
has a simple zero there. To prove the general statement, write
g(z) = (z − z
0
)
N
G(z)
for some
G
with
G
(
z
0
)
6
= 0. Then
1
G(z)
has a Taylor series about
z
0
, and
then the result follows.
(v) z
2
has a double pole at infinity, since
1
ζ
2
has a double pole at ζ = 0.
(vi) e
1/z
has an essential isolated singularity at
z
= 0 because all the
a
n
’s are
non-zero for n ≤ 0.
(vii) sin
1
z
also has an essential isolated singularity at
z
= 0 because (using the
standard Taylor series for
sin
) there are non-zero
a
n
’s for infinitely many
negative n.
(viii) f
(
z
) =
e
z
−1
z
has a removable singularity at
z
= 0, because its Laurent
series is
f(z) = 1 +
1
2!
z +
1
3!
z
2
+ ··· .
By defining
f
(0) = 1, we would remove the singularity and obtain an entire
function.
(ix) f
(
z
) =
sin z
z
is not defined at
z
= 0, but has a removable singularity there;
remove it by setting f(0) = 1.
(x)
A rational function
f
(
z
) =
P (z)
Q(z)
(where
P, Q
are polynomials) has a
singularity at any point
z
0
where
Q
has a zero. Assuming
Q
has a simple
zero, if
P
(
z
0
) = 0 as well, then the singularity is removable by redefining
f(z
0
) =
P
0
(z
0
)
Q
0
(z
0
)
(by L’Hˆopital’s rule).
Near an essential isolated singularity of a function
f
(
z
), it can be shown that
f
takes all possible complex values (except at most one) in any neighbourhood,
however small. For example,
e
1
z
takes all values except zero. We will not prove
this. Even in IB Complex Analysis.