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.