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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