Complex Analysis

by Stein, Elias M.

• ISBN13: 9780691113852
• ISBN10: 0691113858
• Format: Hardcover
• Copyright: 2003-04-07
• Publisher: Ingram Publisher Services, Inc.

Summary

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 Analysiswill 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 whichComplex Analysisis 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

Foreword	vii
Introduction	xv
Chapter 1. Preliminaries to Complex Analysis	1	(31)
1 Complex numbers and the complex plane	1	(7)
1.1 Basic properties	1	(4)
1.2 Convergence	5	(1)
1.3 Sets in the complex plane	5	(3)
2 Functions on the complex plane	8	(10)
2.1 Continuous functions	8	(1)
2.2 Holomorphic functions	8	(6)
2.3 Power series	14	(4)
3 Integration along curves	18	(6)
4 Exercises	24	(8)
Chapter 2. Cauchy's Theorem and Its Applications	32	(39)
1 Goursat's theorem	34	(3)
2 Local existence of primitives and Cauchy's theorem in a disc	37	(4)
3 Evaluation of some integrals	41	(4)
4 Cauchy's integral formulas	45	(8)
5 Further applications	53	(11)
5.1 Morera's theorem	53	(1)
5.2 Sequences of holomorphic functions	53	(2)
5.3 Holomorphic functions defined in terms of integrals	55	(2)
5.4 Schwarz reflection principle	57	(3)
5.5 Runge's approximation theorem	60	(4)
6 Exercises	64	(3)
7 Problems	67	(4)
Chapter 3. Meromorphic Functions and the Logarithm	71	(40)
1 Zeros and poles	72	(4)
2 The residue formula	76	(7)
2.1 Examples	77	(6)
3 Singularities and meromorphic functions	83	(6)
4 The argument principle and applications	89	(4)
5 Homotopies and simply connected domains	93	(4)
6 The complex logarithm	97	(4)
7 Fourier series and harmonic functions	101	(2)
8 Exercises	103	(5)
9 Problems	108	(3)
Chapter 4. The Fourier Transform	111	(23)
1 The class F	113	(1)
2 Action of the Fourier transform on	114	(7)
3 Paley-Wiener theorem	121	(6)
4 Exercises	127	(4)
5 Problems	131	(3)
Chapter 5. Entire Functions	134	(25)
1 Jensen's formula	135	(3)
2 Functions of finite order	138	(2)
3 Infinite products	140	(5)
3.1 Generalities	140	(2)
3.2 Example: the product formula for the sine function	142	(3)
4 Weierstrass infinite products	145	(2)
5 Hadamard's factorization theorem	147	(6)
6 Exercises	153	(3)
7 Problems	156	(3)
Chapter 6. The Gamma and Zeta Functions	159	(22)
1 The gamma function	160	(8)
1.1 Analytic continuation	161	(2)
1.2 Further properties of F	163	(5)
2 The zeta function	168	(6)
2.1 Functional equation and analytic continuation	168	(6)
3 Exercises	174	(5)
4 Problems	179	(2)
Chapter 7. The Zeta Function and Prime Number Theorem	181	(24)
1 Zeros of the zeta function	182	(6)
1.1 Estimates for 1/C(s)	187	(1)
2 Reduction to the functions u and u1	188	(9)
2.1 Proof of the asymptotics for u1	194	(3)
Note on interchanging double sums	197	(2)
3 Exercises	199	(4)
4 Problems	203	(2)
Chapter 8. Conformal Mappings	205	(56)
1 Conformal equivalence and examples	206	(12)
1.1 The disc and upper half-plane	208	(1)
1.2 Further examples	209	(3)
1.3 The Dirichlet problem in a strip	212	(6)
2 The Schwarz lemma; automorphisms of the disc and upper half-plane	218	(6)
2.1 Automorphisms of the disc	219	(2)
2.2 Automorphisms of the upper half-plane	221	(3)
3 The Riemann mapping theorem	224	(7)
3.1 Necessary conditions and statement of the theorem	224	(1)
3.2 Montel's theorem	225	(3)
3.3 Proof of the Riemann mapping theorem	228	(3)
4 Conformal mappings onto polygons	231	(17)
4.1 Some examples	231	(4)
4.2 The Schwarz-Christoffel integral	235	(3)
4.3 Boundary behavior	238	(3)
4.4 The mapping formula	241	(4)
4.5 Return to elliptic integrals	245	(3)
5 Exercises	248	(6)
6 Problems	254	(7)
Chapter 9. An Introduction to Elliptic Functions	261	(22)
1 Elliptic functions	262	(11)
1.1 Liouville's theorems	264	(2)
1.2 The Weierstrass g function	266	(7)
2 The modular character of elliptic functions and Eisenstein series	273	(5)
2.1 Eisenstein series	273	(5)
2.2 Eisenstein series and divisor functions 276
3 Exercises	278	(3)
4 Problems	281	(2)
Chapter 10. Applications of Theta Functions	283	(35)
1 Product formula for the Jacobi theta function	284	(9)
1.1 Further transformation laws	289	(4)
2 Generating functions	293	(3)
3 The theorems about sums of squares	296	(13)
3.1 The two-squares theorem	297	(7)
3.2 The four-squares theorem	304	(5)
4 Exercises	309	(5)
5 Problems	314	(4)
Appendix A: Asymptotics	318	(26)
1 Bessel functions	319	(4)
2 Laplace's method; Stirling's formula	323	(5)
3 The Airy function	328	(6)
4 The partition function	334	(7)
5 Problems	341	(3)
Appendix B: Simple Connectivity and Jordan Curve Theorem	344	(21)
1 Equivalent formulations of simple connectivity	345	(6)
2 The Jordan curve theorem	351	(14)
2.1 Proof of a general form of Cauchy's theorem	361	(4)
Notes and References	365	(4)
Bibliography	369	(4)
Symbol Glossary	373	(2)
Index	375