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## Über dieses Buch

Textbooks, even excellent ones, are a reflection of their times. Form and content of books depend on what the students know already, what they are expected to learn, how the subject matter is regarded in relation to other divisions of mathematics, and even how fashionable the subject matter is. It is thus not surprising that we no longer use such masterpieces as Hurwitz and Courant's Funktionentheorie or Jordan's Cours d'Analyse in our courses. The last two decades have seen a significant change in the techniques used in the theory of functions of one complex variable. The important role played by the inhomogeneous Cauchy-Riemann equation in the current research has led to the reunification, at least in their spirit, of complex analysis in one and in several variables. We say reunification since we think that Weierstrass, Poincare, and others (in contrast to many of our students) did not consider them to be entirely separate subjects. Indeed, not only complex analysis in several variables, but also number theory, harmonic analysis, and other branches of mathematics, both pure and applied, have required a reconsidera­ tion of analytic continuation, ordinary differential equations in the complex domain, asymptotic analysis, iteration of holomorphic functions, and many other subjects from the classic theory of functions of one complex variable. This ongoing reconsideration led us to think that a textbook incorporating some of these new perspectives and techniques had to be written.

## Inhaltsverzeichnis

### Chapter 1. Topology of the Complex Plane and Holomorphic Functions

Abstract
The complex plane ℂ coincides with ℝ2 by the usual identification of a complex number z = x + iy, x = Re z, y = Im z, with the vector (x, y). As such it has two vector space structures, one as a two-dimensional vector space over ℝ and the other as a one-dimensional vector space over ℂ. The relations between them lead to the classical Cauchy-Riemann equations.
Carlos A. Berenstein, Roger Gay

### Chapter 2. Analytic Properties of Holomorphic Functions

Abstract
At the end of Chapter 1 we introduced the holomorphic functions, that is, those functions fC1(Ω) that satisfy the Cauchy-Riemann differential equation $$\frac{{\partial f}}{{\partial \bar z}} = 0$$ throughout an open set Ω ⊆ ℂ. As an immediate consequence of the topological tools developed in that chapter we found that the holomorphic functions enjoyed the following remarkable property (Cauchy’s theorem 1.1 1.4).
Carlos A. Berenstein, Roger Gay

### Chapter 3. The -Equation

Abstract
It is in this chapter that the difference between our textbook and more classical ones appears markedly. As stated in the preface, we have attempted to use, as systematically as possible, the inhomogeneous Cauchy-Riemann equation $$\frac{{\partial f}}{{\partial \bar z}} = g$$ to study holomorphic functions (also called $$\bar \partial$$-equation). The reader should note the irony here. To better comprehend the solutions of the homogeneous equation $$\frac{{\partial f}}{{\partial \bar z}} = 0$$ one is forced to study a more complex object! Our presentation owes much to Hörmander’s beautiful treatise on several complex variables [Ho1].
Carlos A. Berenstein, Roger Gay

### Chapter 4. Harmonic and Subharmonic Functions

Abstract
A large number of properties of hololomorphic functions (maximum principle, Schwarz’s lemma, convexity properties, etc.) still hold for a much larger class of functions. It is the class of subharmonic functions (see Definition 4.4.1). The relation between these two classes of functions is given by the fact that if f is a holomorphic function, then log | f | is a subharmonic function.
Carlos A. Berenstein, Roger Gay

### Chapter 5. Analytic Continuation and Singularities

Abstract
When we say “given a holomorphic function in an open set Ω,” we are already making a choice of the domain of the function Sometimes it is evident that the function is in fact the restriction to Ω of a holomorphic function defined on a larger open set. The obvious example of a removable isolated singularity comes to mind. Another example occurs when we define the function by a power series expansion, for instance, for $$f(z) = \sum\limits_{n \geqslant 0} {{z^n}}$$ in B(0, 1), we can sum the series and find that the function z↦(1 — z)-1, holomorphic in ℂ\{1}, extends the function f to this larger open set.
Carlos A. Berenstein, Roger Gay

### Backmatter

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