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2001 | Buch | 2. Auflage

Elementary Theory of Holomorphic Functions Kapitel lesen Erstes Kapitel lesen

Complex Analysis in One Variable

verfasst von: Raghavan Narasimhan, Yves Nievergelt

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

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Über dieses Buch

The original edition of this book has been out of print for some years. The appear­ ance of the present second edition owes much to the initiative of Yves Nievergelt at Eastern Washington University, and the support of Ann Kostant, Mathematics Editor at Birkhauser. Since the book was first published, several people have remarked on the absence of exercises and expressed the opinion that the book would have been more useful had exercises been included. In 1997, Yves Nievergelt informed me that, for a decade, he had regularly taught a course at Eastern Washington based on the book, and that he had systematically compiled exercises for his course. He kindly put his work at my disposal. Thus, the present edition appears in two parts. The first is essentially just a reprint of the original edition. I have corrected the misprints of which I have become aware (including those pointed out to me by others), and have made a small number of other minor changes.

Inhaltsverzeichnis

Frontmatter

Complex Analysis in One Variable

Frontmatter
Chapter 1. Elementary Theory of Holomorphic Functions
Abstract
In this chapter, we shall develop the classical theory of holomorphic functions. The Looman—Menchoff theorem, proved in §1.6, is less standard than the rest of the material.
Raghavan Narasimhan, Yves Nievergelt
Chapter 2. Covering Spaces and the Monodromy Theorem
Abstract
We shall develop the results of this chapter in the context of manifolds (Definition 1 in §2.1 below) although these results, and most of their proofs, remain valid for more general spaces. This is done to keep the statements relatively simple, and manifolds are ample for the applications we have in mind.
Raghavan Narasimhan, Yves Nievergelt
Chapter 3. The Winding Number and the Residue Theorem
Abstract
The homotopy form of Cauchy’s theorem enables one to calculate many integrals of the form ∫ γ À; dz, whereÀ; is meromorphic and γ is a closed piecewise differentiable curve (it being assumed that the poles ofÀ; do not lie on Im(γ)). Formulae enabling one to do this include the so-called Cauchy formula (see §2, Theorem 2). It is, however, necessary to have some topological information about the location of the poles relative to γ. (To phrase it very vaguely, we must know how many times γ winds around a.) We begin with this topological material.
Raghavan Narasimhan, Yves Nievergelt
Chapter 4. Picard’s Theorem
Abstract
In this chapter, we shall prove the so-called “big” theorem of Picard which asserts that a holomorphic function with an (isolated) essential singularity assumes every value with at most one exception in any neighborhood of that singularity.
Raghavan Narasimhan, Yves Nievergelt
Chapter 5. The Inhomogeneous Cauchy-Riemann Equation and Runge’s Theorem
Abstract
Holomorphic functions are characterized by the equation ∂É/∂z = 0. In this chapter, we shall study the equation ∂É/∂̄z = g when g has compact support. We shall obtain an explicit solution which leads to a variant of the Cauchy integral formula. This variant can often be used instead of the usual Cauchy formula, and has the advantage of not involving winding numbers. We shall illustrate this principle with a variant of the argument principle and a proof of the Runge theorem.
Raghavan Narasimhan, Yves Nievergelt
Chapter 6. Applications of Runge’s Theorem
Abstract
This chapter is devoted to various theorems which can be proved using Runge’s theorem: the existence of functions with prescribed zeros or poles, a “cohomological” version of Cauchy’s theorem, and related theorems. The last section concerns itself with Η(Ω) as a ring (or ℂ-algebra).
Raghavan Narasimhan, Yves Nievergelt
Chapter 7. The Riemann Mapping Theorem and Simple Connectedness in the Plane
Abstract
In this chapter, we shall prove that any simply connected open set in ℂ, which is not all of ℂ, is analytically isomorphic to the unit disc D = }z ∊ ℂz < 1}. The proof will also enable us to characterize simple connectedness in several ways.
Raghavan Narasimhan, Yves Nievergelt
Chapter 8. Functions of Several Complex Variables
Abstract
In this chapter, we shall define holomorphic functions of several complex variables. The essentially local theory given in Chapter 1, §§3, 4 extends to these functions with little effort. We shall then prove two theorems which show that the behavior of functions of n complex variables, with n > 1, is, in some ways, radically different from that of functions of one variable.
Raghavan Narasimhan, Yves Nievergelt
Chapter 9. Compact Riemann Surfaces
Abstract
In this chapter, we introduce Riemann surfaces and prove an important theorem which asserts that meromorphic functions on a compact Riemann surface form an algebraic function field in one variable (see §6). The chapter is meant to serve as an introduction to some tools which have proved to be very useful in several branches of mathematics, in particular, in several complex variables and algebraic geometry.
Raghavan Narasimhan, Yves Nievergelt
Chapter 10. The Corona Theorem
Abstract
We saw in Chapter 6 that if Ω is open in ℂ andÀ;1,... ,À; n ∈ ℌ(Ω) and have no common zeros in Ω, then there exist g 1,... , g n ∊ Η(Ω) such that ∑ g i À; 1 ≡ 1.
Raghavan Narasimhan, Yves Nievergelt
Chapter 11. Subharmonic Functions and the Dirichlet Problem
Abstract
In this chapter we introduce and study subharmonic functions and use them to solve the Dirichlet problem for harmonic functions (on reasonable domains). We shall indicate some other applications of these functions at the end of the chapter.
Raghavan Narasimhan, Yves Nievergelt
Backmatter

Exercises

Frontmatter
Introduction
Abstract
Many of the following exercises were contributed by Narasimhan, Kevin Corlette, and Madhav Vithal Nori, from courses that they taught at the University of Chicago. Several versions of other exercises have accompanied Narasimhan’s text for a decade in a first-year, two-trimester graduate course (following or concurrent with a trimester of topology) taught at Eastern Washington University. The collection of exercises proposed here addresses an audience of students who have demonstrated abilities for graduate studies in mathematics and yet may pursue any of the following different goals:
  • doctoral work in mathematics,
  • teaching careers in Community Colleges or high schools, and
  • technical positions in government or industry.
Yves Nievergelt
Chapter 0. Review of Complex Numbers
Abstract
The following exercises summarize the definition and elementary algebraic features of the complex numbers.
Raghavan Narasimhan, Yves Nievergelt
Chapter 1. Elementary Theory of Holomorphic Functions
Abstract
The following exercises extend differential calculus to complex analysis.
Raghavan Narasimhan, Yves Nievergelt
Chapter 2. Covering Spaces and the Monodromy Theorem
Abstract
The following exercises provide some practice with manifolds that arise frequently in mathematics. The exercises for Chapter 9 contain other examples amenable to the methods from Chapter 2.
Raghavan Narasimhan, Yves Nievergelt
Chapter 3. The Winding Number and the Residue Theorem
Abstract
Exercise 182. Prove that for each compact subset K ⊂ ℂ the complement ℂ \ K has exactly one unbounded connected component.
Raghavan Narasimhan, Yves Nievergelt
Chapter 4. Picard’s Theorem
Abstract
Exercise 220. Prove that ifÀ; and g are entire and e É + e g = 1, thenÀ; and g are constant.
Raghavan Narasimhan, Yves Nievergelt
Chapter 5. The Inhomogeneous Cauchy-Riemann Equation and Runge’s Theorem
Abstract
These exercises focus on details of the proof of the existence of partitions of unity.
Raghavan Narasimhan, Yves Nievergelt
Chapter 6. Applications of Runge’s Theorem
Abstract
The following exercise verifies a step in the proof of Theorem 1 in Section 1 of Chapter 6.
Raghavan Narasimhan, Yves Nievergelt
Chapter 7. The Riemann Mapping Theorem and Simple Connectedness in the Plane
Abstract
Exercise 293. Determine the types of fractional linear transformations that map the upper half plane onto the unit disc, with the real axis onto the unit circle.
Raghavan Narasimhan, Yves Nievergelt
Chapter 8. Functions of Several Complex Variables
Abstract
Exercise 311. Consider a power series
$$ \sum\limits_{m \in n} {\sum\limits_{n \in n} {{c_{m,n}}{z^m}{w^n}} } $$
that converges to a limitÀ;(z, ω) for all z ℂ and ω ∈ ℂ for which |z| + |ω| < R.
Raghavan Narasimhan, Yves Nievergelt
Chapter 9. Compact Riemann Surfaces
Abstract
Exercise 322. For each τ ∈ ℂ with ℑm(τ) > 0, define the lattice ⋀ τ := ℤ × τℤ ⊂ ℂ, and define the complex torus X τ := ℂ/⋀ τ . For two such complex numbers τ 1, τ 2 ∈ ℂ with ℑm(τ 1) > 0 and ℑm(τ 2) > 0, assume that there exist a holomorphic isomorphism \( f:{X_{{\tau _1}}} \to {X_{{\tau _2}}} \) and an entire function g : ℂ → ℂ such that the following diagram commutes:
$$ {p_1}\matrix{ c & {\buildrel g \over \longrightarrow } & c \cr \downarrow & {} & \downarrow \cr {{X_{{\tau _1}}}} & {\mathrel{\mathop{\kern0pt\longrightarrow} \limits_f} } & {{X_{{\tau _2}}}} \cr } {p_2} $$
where each \( {p_k}:c \to {X_{{\tau _k}}} = c/{\Lambda _{{\tau _k}}} \) is the canonical projection.
Raghavan Narasimhan, Yves Nievergelt
Chapter 10. The Corona Theorem
Abstract
Exercise 348. This exercise outlines another derivation of Poisson’s integral formula, based on Cauchy’s integral formula. To this end, assume first thatÀ; ∈ ℌ[D(0, R)] for some R > 1.
Raghavan Narasimhan, Yves Nievergelt
Chapter 11. Subharmonic Functions and the Dirichlet Problem
Abstract
Exercise 353. Consider an open set Ω ⊆ ℂ and a real-valued harmonic function u on Ω.
Raghavan Narasimhan, Yves Nievergelt
Backmatter
Metadaten
Titel
Complex Analysis in One Variable
verfasst von
Raghavan Narasimhan
Yves Nievergelt
Copyright-Jahr
2001
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-0175-5
Print ISBN
978-1-4612-6647-1
DOI
https://doi.org/10.1007/978-1-4612-0175-5