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
− ··· .