2Contour integration
IB Complex Analysis
2.4 Taylor’s theorem
When we first met Taylor series, we were happy, since we can express anything
as a power series. However, we soon realized this is just a fantasy — the Taylor
series of a real function need not be equal to the function itself. For example,
the function
f
(
x
) =
e
−x
−2
has vanishing Taylor series at 0, but does not vanish
in any neighbourhood of 0. What we do have is Taylor’s theorem, which gives
you an expression for what the remainder is if we truncate our series, but is
otherwise completely useless.
In the world of complex analysis, we are happy once again. Every holomorphic
function can be given by its Taylor series.
Theorem
(Taylor’s theorem)
.
Let
f
:
B
(
a, r
)
→ C
be holomorphic. Then
f
has
a convergent power series representation
f(z) =
∞
X
n=0
c
n
(z −a)
n
on B(a, r). Moreover,
c
n
=
f
(n)
(a)
n!
=
1
2πi
Z
∂B(a,ρ)
f(z)
(z −a)
n+1
dz
for any 0 < ρ < r.
Note that the very statement of the theorem already implies any holomorphic
function has to be infinitely differentiable. This is a good world.
Proof. We’ll use Cauchy’s integral formula. If |w −a| < ρ < r, then
f(w) =
1
2πi
Z
∂B(a,ρ)
f(z)
z −w
dz.
Now (cf. the first proof of the Cauchy integral formula), we note that
1
z −w
=
1
(z −a)
1 −
w−a
z−a
=
n
X
n=0
(w − a)
n
(z −a)
n+1
.
This series is uniformly convergent everywhere on the
ρ
disk, including its
boundary. By uniform convergence, we can exchange integration and summation
to get
f(w) =
∞
X
n=0
1
2πi
Z
∂B(a,ρ)
f(z)
(z −a)
n+1
dz
!
(w − a)
n
=
∞
X
n=0
c
n
(w − a)
n
.
Since
c
n
does not depend on
w
, this is a genuine power series representation,
and this is valid on any disk B(a, ρ) ⊆ B(a, r).
Then the formula for
c
n
in terms of the derivative comes for free since that’s
the formula for the derivative of a power series.
This tells us every holomorphic function behaves like a power series. In
particular, we do not get weird things like
e
−x
−2
on
R
that have a trivial Taylor
series expansion, but is itself non-trivial. Similarly, we know that there are no
“bump functions” on
C
that are non-zero only on a compact set (since power
series don’t behave like that). Of course, we already knew that from Liouville’s
theorem.
Corollary.
If
f
:
B
(
a, r
)
→ C
is holomorphic on a disc, then
f
is infinitely
differentiable on the disc.
Proof.
Complex power series are infinitely differentiable (and
f
had better be
infinitely differentiable for us to write down the formula for
c
n
in terms of
f
(n)
).
This justifies our claim from the very beginning that
Re
(
f
) and
Im
(
f
) are
harmonic functions if f is holomorphic.
Corollary.
If
f
:
U → C
is a complex-valued function, then
f
=
u
+
iv
is
holomorphic at
p ∈ U
if and only if
u, v
satisfy the Cauchy–Riemann equations,
and that u
x
, u
y
, v
x
, v
y
are continuous in a neighbourhood of p.
Proof.
If
u
x
, u
y
, v
x
, v
y
exist and are continuous in an open neighbourhood of
p
,
then
u
and
v
are differentiable as functions
R
2
→ R
2
at
p
, and then we proved
that the Cauchy–Riemann equations imply differentiability at each point in the
neighbourhood of p. So f is differentiable at a neighbourhood of p.
On the other hand, if
f
is holomorphic, then it is infinitely differentiable. In
particular,
f
0
(
z
) is also holomorphic. So
u
x
, u
y
, v
x
, v
y
are differentiable, hence
continuous.
We also get the following (partial) converse to Cauchy’s theorem.
Corollary
(Morera’s theorem)
.
Let
U ⊆ C
be a domain. Let
f
:
U → C
be
continuous such that
Z
γ
f(z) dz = 0
for all piecewise-C
1
closed curves γ ∈ U . Then f is holomorphic on U .
Proof.
We have previously shown that the condition implies that
f
has an
antiderivative
F
:
U → C
, i.e.
F
is a holomorphic function such that
F
0
=
f
.
But F is infinitely differentiable. So f must be holomorphic.
Recall that Cauchy’s theorem required
U
to be sufficiently nice, e.g. being
star-shaped or just simply-connected. However, Morera’s theorem does not. It
just requires that
U
is a domain. This is since holomorphicity is a local property,
while vanishing on closed curves is a global result. Cauchy’s theorem gets us
from a local property to a global property, and hence we need to assume more
about what the “globe” looks like. On the other hand, passing from a global
property to a local one does not. Hence we have this asymmetry.
Corollary.
Let
U ⊆ C
be a domain,
f
n
;
U → C
be a holomorphic function. If
f
n
→ f uniformly, then f is in fact holomorphic, and
f
0
(z) = lim
n
f
0
n
(z).
Proof. Given a piecewise C
1
path γ, uniformity of convergence says
Z
γ
f
n
(z) dz →
Z
γ
f(z) dz
uniformly. Since
f
being holomorphic is a local condition, so we fix
p ∈ U
and
work in some small, convex disc
B
(
p, ε
)
⊆ U
. Then for any curve
γ
inside this
disk, we have
Z
γ
f
n
(z) dz = 0.
Hence we also have
R
γ
f
(
z
) d
z
= 0. Since this is true for all curves, we conclude
f
is holomorphic inside
B
(
p, ε
) by Morera’s theorem. Since
p
was arbitrary, we
know f is holomorphic.
We know the derivative of the limit is the limit of the derivative since we can
express f
0
(a) in terms of the integral of
f(z)
(z−a)
2
, as in Taylor’s theorem.
There is a lot of passing between knowledge of integrals and knowledge of
holomorphicity all the time, as we can see in these few results. These few sections
are in some sense the heart of the course, where we start from Cauchy’s theorem
and Cauchy’s integral formula, and derive all the other amazing consequences.