2
1. Selected Problems in One Complex Variable
1.1 Preliminaries
The complex plane will be denoted by C, while complex n-spaee, the Carte-
sian product of n copies of C, will be denoted by C n . The open disc of
radius r 0 centered at a G C will be denoted A (a, r), while the closed disc
with this radius and center will be denoted A(a, r). If U is an open set in
C and / a complex valued function defined on [/, then / is holomorphic if
its complex derivative
/'(*)
= H m
f{z)
~
I{W)
w-^z
Z W
exists for each z G U. We will denote the space of all holomorphic functions
on U by H(U).
We assume that the reader is familiar with the basic properties of holo-
morphic functions of a single variable as presented in standard texts (e.g.
[R]) in the subject: A holomorphic function on U has a convergent power
series expansion in a neighborhood of each point of [/; a differentiate func-
tion is holomorphic if and only if it satisfies the Cauchy-Riemann equations;
the space H(U) is an algebra over the complex field under the operations of
pointwise addition, multiplication, and scalar multiplication; if a sequence
{fn} in 1~L{U) converges uniformly on each compact subset of [/, then the
limit function is also holomorphic on [/; holomorphic functions satisfy the
Cauchy integral theorem and formula, the identity theorem, and the maxi-
mum modulus theorem.
A function on an open set U in C
n
is holomorphic if it is holomorphic
in each variable separately (in the next chapter we shall prove that this is
equivalent to the existence of local multi-variable power series expansions
of the function). In the n variable case, we shall also denote the space of
holomorphic functions on U by H(U). The space of continuous functions
on a topological space U will be denoted by C{U)1 the space of n times
continuously differentiate functions on an open set U in a Euclidean space
by
Cn(?7),
and the space of infinitely differentiate functions on U by C°°(U).
As usual, the Euclidean norm of a point z (z\1 Z2,..., zn) G C n is defined
by|kll = (N
2
+
M2
+ --- +
kn|2)1/2.
1.2 A Simple Proble m
Here we introduce a problem which is easy to state, but not so easy to solve.
Its solution illustrates, in a relatively simple setting, many of the issues we
will have to deal with later in this text.
Let U be an open set in C. Since H(U) is an algebra, it is natural to
try to find all of its maximal ideals. It is easy to see that each point A G U
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