Complex Analysis (Princeton Lectures in Analysis, Volume II)

✍ Scribed by Elias M. Stein, Rami Shakarchi

• Publisher: Princeton University Press
• Year: 2003
• Tongue: English
• Leaves: 398
• Series: Princeton Lectures in Analysis, 2
• Edition: 1
• Category: Library

✦ Synopsis

With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle.


With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory.


Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences.


The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the far-reaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to in-depth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.

✦ Table of Contents

Book II
Cover
Half-Title
Title
Copyright
Authors’ Dedications
Foreword
Contents
Introduction
Chapter 1. Preliminaries to Complex Analysis
1.1 Complex numbers and the complex plane
1.1.1 Basic properties
1.1.2 Convergence
1.1.3 Sets in the complex plane
1.2 Functions on the complex plane
1.2.1 Continuous functions
1.2.2 Holomorphic functions
Complex-valued functions as mappings
1.2.3 Power series
1.3 Integration along curves
1.4 Exercises
Chapter 2. Cauchy’s Theorem and Its Applications
2.1 Goursat’s theorem
2.2 Local existence of primitives and Cauchy’s theorem in a disc
2.3 Evaluation of some integrals
2.4 Cauchy’s integral formulas
2.5 Further applications
2.5.1 Morera’s theorem
2.5.2 Sequences of holomorphic functions
2.5.3 Holomorphic functions defined in terms of integrals
2.5.4 Schwarz reflection principle
2.5.5 Runge’s approximation theorem
2.6 Exercises
2.7 Problems
Chapter 3. Meromorphic Functions and the Logarithm
3.1 Zeros and poles
3.2 The residue formula
3.2.1 Examples
3.3 Singularities and meromorphic functions
The Riemann sphere
3.4 The argument principle and applications
3.5 Homotopies and simply connected domains
3.6 The complex logarithm
3.7 Fourier series and harmonic functions
3.8 Exercises
3.9 Problems
Chapter 4. The Fourier Transform
4.1 The class F
4.2 Action of the Fourier transform on F
4.3 Paley-Wiener theorem
4.4 Exercises
4.5 Problems
Chapter 5. Entire Functions
5.1 Jensen’s formula
5.2 Functions of finite order
5.3 Infinite products
5.3.1 Generalities
5.3.2 Example: the product formula for the sine function
5.4 Weierstrass infinite products
5.5 Hadamard’s factorization theorem
Main lemmas
Proof of Hadamard’s theorem
5.6 Exercises
5.7 Problems
Chapter 6. The Gamma and Zeta Functions
6.1 The gamma function
6.1.1 Analytic continuation
6.1.2 Further properties of Γ
6.2 The zeta function
6.2.1 Functional equation and analytic continuation
6.3 Exercises
6.4 Problems
Chapter 7. The Zeta Function and Prime Number Theorem
7.1 Zeros of the zeta function
7.1.1 Estimates for 1/ζ(s)
7.2 Reduction to the functions ψ and ψ1
7.2.1 Proof of the asymptotics for ψ1
Note on interchanging double sums
7.3 Exercises
7.4 Problems
Chapter 8. Conformal Mappings
8.1 Conformal equivalence and examples
8.1.1 The disc and upper half-plane
8.1.2 Further examples
8.1.3 The Dirichlet problem in a strip
Remarks about the Dirichlet problem
8.2 The Schwarz lemma; automorphisms of the disc and upper half-plane
8.2.1 Automorphisms of the disc
8.2.2 Automorphisms of the upper half-plane
8.3 The Riemann mapping theorem
8.3.1 Necessary conditions and statement of the theorem
8.3.2 Montel’s theorem
8.3.3 Proof of the Riemann mapping theorem
8.4 Conformal mappings onto polygons
8.4.1 Some examples
8.4.2 The Schwarz-Christoffel integral
8.4.3 Boundary behavior
8.4.4 The mapping formula
8.4.5 Return to elliptic integrals
8.5 Exercises
8.6 Problems
Chapter 9. An Introduction to Elliptic Functions
9.1 Elliptic functions
9.1.1 Liouville’s theorems
9.1.2 The Weierstrass ℘ function
An elliptic function of order two
Properties of ℘
9.2 The modular character of elliptic functions and Eisenstein series
9.2.1 Eisenstein series
9.2.2 Eisenstein series and divisor functions
9.3 Exercises
9.4 Problems
Chapter 10. Applications of Theta Functions
10.1 Product formula for the Jacobi theta function
10.1.1 Further transformation laws
10.2 Generating functions
10.3 The theorems about sums of squares
10.3.1 The two-squares theorem
10.3.2 The four-squares theorem
Statement of the theorem
10.4 Exercises
10.5 Problems
Appendix A: Asymptotics
A.1 Bessel functions
A.2 Laplace’s method; Stirling’s formula
A.3 The Airy function
A.4 The partition function
A.5 Problems
Appendix B: Simple Connectivity and Jordan Curve Theorem
B.1 Equivalent formulations of simple connectivity
Winding numbers
B.2 The Jordan curve theorem
Proof of Theorem 2.1
Proof of Theorem 2.2
B.2.1 Proof of a general form of Cauchy’s theorem
Notes and References
Bibliography
Symbol Glossary
Index