F
Complex Analysis
In this appendix, we collect a few basic definitions and facts from complex
analysis. Complex analysis is a vast and beautiful subject. Although we make
only limited use of complex analysis in this volume, there is a rich interaction
between harmonic analysis and complex analysis, some of which can be seen in
the texts by Dym and McKean [DM72] and Young [You01]. Some basic texts
on complex analysis include Conway [Con78], Marsden and Hoffman [MH87],
and Stein and Shakarchi [SS03b].
F.1 Analytic Functions
Complex analysis is concerned with functions that map the complex plane to
itself. An analytic function is one that has a complex derivative. This is a very
strong requirement — no matter what “path” to the origin that we take, the
limit in the definition of the derivative must exist.
Definition F.1 (Analytic Function).
Let Ω
⊆
C
be open. Then a function
f
: Ω
→
C
is
analytic
or
holomorphic
on Ω if the limit
f
′
(
z
) =
lim
h
→
0
f
(
z
+
h
)
−
f
(
z
)
h
(F.1)
exists for all
z
∈
Ω
.
The function
f
′
is called the
derivative
or
complex deriva
tive
of
f.
If
f
:
C
→
C
is analytic on
C
,
then we say that
f
is
entire
.
Note that the variable
h
in equation (F.1) is a complex variable. The
meaning of the limit is that for every
ε >
0
,
there exists a
δ >
0 such that
0
<

h

< δ, z
+
h
∈
Ω
=
⇒
vextendsingle
vextendsingle
vextendsingle
f
′
(
z
)
−
f
(
z
+
h
)
−
f
(
z
)
h
vextendsingle
vextendsingle
vextendsingle
< ε.
For example, any polynomial
p
(
z
) =
a
0
+
a
1
z
+
· · ·
+
a
n
z
n
is entire, as is
the exponential function
e
z
.
The function
f
(
z
) = 1
/z
is analytic on
C
\{
0
}
.
If
r >
0
,
then we set
r
z
=
e
z
ln
r
.
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F Complex Analysis
There are many equivalent formulations of analyticity, which we will not
present. For us, the important fact is that analytic functions are very highly
constrained. Some of the properties of analytic functions are laid out in the
next theorem. In particular, an analytic function is uniquely determined by
its values on any infinite set that has an accumulation point.
Theorem F.2 (Properties of Analytic Functions).
If
f
is analytic on
an open set
Ω
⊆
C
,
then the following statements hold.
(a)
f
′
is analytic on
Ω
.
(b)
f
has infinitely many complex derivatives on
Ω
.
(c)
Let
S
be any infinite subset of
Ω
that has an accumulation point in
Ω
.
If
g
is also analytic on
Ω
and
f
(
z
) =
g
(
z
)
for all
z
∈
S,
then
f
(
z
) =
g
(
z
)
for all
z
∈
Ω
.
We also have the following two important properties of analytic functions.
Theorem F.3 (Liouville’s Theorem).
If
f
is both bounded and analytic
on all of
C
,
then
f
is constant.
Theorem F.4 (Maximum Modulus Principle).
Let
Ω
be a bounded,
open, and connected subset of
C
.
Suppose that
f
is analytic on
Ω
and continu
ous on
Ω
,
and let
M
= sup
z
∈
∂
Ω

f
(
z
)

be the maximum of

f

on the boundary
of
Ω
.
Then

f
(
z
)
 ≤
M
for all
z
∈
Ω
.
Moreover, if

f
(
z
)

=
M
for some
z
∈
Ω
then
f
is constant.
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 Fall '09
 HEIL
 Taylor Series, The Land, analytic functions, Properties of Analytic functions

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