1Complex differentiation
IB Complex Analysis
1.3 Power series
Some of our favorite functions are all power series, including polynomials (which
are degenerate power series in some sense), exponential functions and trigono-
metric functions. We will later show that all holomorphic functions are (locally)
given by power series, but without knowing this fact, we shall now study some
of the basic properties of power series.
It turns out power series are nice. The key property is that power series
are infinitely differentiable (as long as it converges), and the derivative is given
pointwise.
We begin by recalling some facts about convergence from IB Analysis II.
Definition
(Uniform convergence)
.
A sequence (
f
n
) of functions converge
uniformly to
f
if for all
ε >
0, there is some
N
such that
n > N
implies
|f
n
(z) − f(z)| < ε for all z.
Proposition. The uniform limit of continuous functions is continuous.
Proposition
(Weierstrass M-test)
.
For a sequence of functions
f
n
, if we can
find (
M
n
)
⊆ R
>0
such that
|f
n
(
x
)
| < M
n
for all
x
in the domain, then
P
M
n
converges implies
P
f
n
(x) converges uniformly on the domain.
Proposition.
Given any constants
{c
n
}
n≥0
⊆ C
, there is a unique
R ∈
[0
, ∞
]
such that the series
z 7→
P
∞
n=0
c
n
(
z − a
)
n
converges absolutely if
|z − a| < R
and diverges if
|z −a| > R
. Moreover, if 0
< r < R
, then the series converges
uniformly on {z : |z −a| < r}. This R is known as the radius of convergence.
So while we don’t necessarily get uniform convergence on the whole domain,
we get uniform convergence on all compact subsets of the domain.
We are now going to look at power series. They will serve as examples, and
as we will see later, universal examples, of holomorphic functions. The most
important result we need is the following result about their differentiability.
Theorem. Let
f(z) =
∞
X
n=0
c
n
(z − a)
n
be a power series with radius of convergence R > 0. Then
(i) f is holomorphic on B(a; R) = {z : |z − a| < R}.
(ii) f
0
(z) =
P
nc
n
(z − 1)
n−1
, which also has radius of convergence R.
(iii) Therefore f is infinitely complex differentiable on B(a; R). Furthermore,
c
n
=
f
(n)
(a)
n!
.
Proof.
Without loss of generality, take
a
= 0. The third part obviously follows
from the previous two, and we will prove the first two parts simultaneously. We
would like to first prove that the derivative series has radius of convergence
R
,
so that we can freely happily manipulate it.
Certainly, we have
|nc
n
| ≥ |c
n
|
. So by comparison to the series for
f
, we can
see that the radius of convergence of
P
nc
n
z
n−1
is at most
R
. But if
|z| < ρ < R
,
then we can see
|nc
n
z
n−1
|
|c
n
ρ
n−1
|
= n
z
ρ
n−1
→ 0
as
n → ∞
. So by comparison to
P
c
n
ρ
n−1
, which converges, we see that the
radius of convergence of
P
nc
n
z
n−1
is at least
ρ
. So the radius of convergence
must be exactly R.
Now we want to show
f
really is differentiable with that derivative. Pick
z, w such that |z|, |w| ≤ ρ for some ρ < R as before.
Define a new function
ϕ(z, w) =
∞
X
n=1
c
n
n−1
X
j=0
z
j
w
n−1−j
.
Noting
c
n
n−1
X
j=0
z
j
w
n−1−j
≤ n|c
n
|ρ
n
,
we know the series defining
ϕ
converges uniformly on
{|z| ≤ ρ, |w| < ρ}
, and
hence to a continuous limit.
If z 6= w, then using the formula for the (finite) geometric series, we know
ϕ(z, w) =
∞
X
n=1
c
n
z
n
− w
n
z − w
=
f(z) − f (w)
z − w
.
On the other hand, if z = w, then
ϕ(z, z) =
∞
X
n=1
c
n
nz
n−1
.
Since ϕ is continuous, we know
lim
w→z
f(z) − f (w)
z − w
→
∞
X
n=1
c
n
nz
n−1
.
So f
0
(z) = ϕ(z, z) as claimed. Then (iii) follows from (i) and (ii) directly.
Corollary. Given a power series
f(z) =
X
n≥0
c
n
(z − a)
n
with radius of convergence
R >
0, and given 0
< ε < R
, if
f
vanishes on
B
(
a, ε
),
then f vanishes identically.
Proof.
If
f
vanishes on
B
(
a, ε
), then all its derivatives vanish, and hence the
coefficients all vanish. So it is identically zero.
This is obviously true, but will come up useful some time later.
It is might be useful to have an explicit expression for
R
. For example, by
IA Analysis I, we know
R = sup{r ≥ 0 : |c
n
|r
n
→ 0 as n → ∞}
=
1
lim sup
n
p
|c
n
|
.
But we probably won’t need these.