3Laurent series and singularities

IB Complex Methods

3.2 Zeros

Recall that for a polynomial

p

(

z

), we can talk about the order of its zero at

z

=

a

by looking at the largest power of (

z − a

) dividing

p

. A priori, it is not

clear how we can do this for general functions. However, given that everything

is a Taylor series, we know how to do this for holomorphic functions.

Definition

(Zeros)

.

The zeros of an analytic function

f

(

Z

) are the points

z

0

where

f

(

z

0

) = 0. A zero is of order

N

if in its Taylor series

P

∞

n=0

a

n

(

z − z

0

)

n

,

the first non-zero coefficient is a

N

.

Alternatively, it is of order

N

if 0 =

f

(

z

0

) =

f

0

(

z

0

) =

···

=

f

(N−1)

, but

f

(N)

(z

0

) 6= 0.

Definition (Simple zero). A zero of order one is called a simple zero.

Example. z

3

+

iz

2

+

z

+

i

= (

z − i

)(

z

+

i

)

2

has a simple zero at

z

=

i

and a

zero of order 2 at z = −i.

Example. sinh z

has zeros where

1

2

(

e

z

− e

−z

) = 0, i.e.

e

2z

= 1, i.e.

z

=

nπi

,

where n ∈ Z. The zeros are all simple, since cosh nπi = cos nπ 6= 0.

Example.

Since

sinh z

has a simple zero at

z

=

πi

, we know

sinh

3

z

has a zero

of order 3 there. This is since the first term of the Taylor series of

sinh z

about

z

=

πi

has order 1, and hence the first term of the Taylor series of

sinh

3

z

has

order 3.

We can also find the Taylor series about πi by writing ζ = z − πi:

sinh

3

z = [sinh(ζ + πi)]

3

= [−sinh ζ]

3

= −

ζ +

1

3!

+ ···

3

= −ζ

3

−

1

2

ζ

5

− ···

= −(z − πi)

3

−

1

2

(z − πi)

5

− ··· .