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.

Foreword vii
Introduction xv

Chapter 1. Preliminaries to Complex Analysis 1
1 Complex numbers and the complex plane 1
1.1 Basic properties 1
1.2 Convergence 5
1.3 Sets in the complex plane 5
2 Functions on the complex plane 8
2.1 Continuous functions 8
2.2 Holomorphic functions 8
2.3 Power series 14
3 Integration along curves 18
4 Exercises 24

Chapter 2. Cauchy's Theorem and Its Applications 32
1 Goursat's theorem 34
2 Local existence of primitives and Cauchy's theorem in a disc 37
3 Evaluation of some integrals 41
4 Cauchy's integral formulas 45
5 Further applications 53
5.1 Morera's theorem 53
5.2 Sequences of holomorphic functions 53
5.3 Holomorphic functions defined in terms of integrals 55
5.4 Schwarz reflection principle 57
5.5 Runge's approximation theorem 60
6 Exercises 64
7 Problems 67

Chapter 3. Meromorphic Functions and the Logarithm 71
1 Zeros and poles 72
2 The residue formula 76
2.1 Examples 77
3 Singularities and meromorphic functions 83
4 The argument principle and applications 89
5 Homotopies and simply connected domains 93
6 The complex logarithm 97
7 Fourier series and harmonic functions 101
8 Exercises 103
9 Problems 108

Chapter 4. The Fourier Transform 111
1 The class F 113
2 Action of the Fourier transform on F 114
3 Paley-Wiener theorem 121
4 Exercises 127
5 Problems 131

Chapter 5. Entire Functions 134
1 Jensen's formula 135
2 Functions of finite order 138
3 Infinite products 140
3.1 Generalities 140
3.2 Example: the product formula for the sine function 142
4 Weierstrass infinite products 145
5 Hadamard's factorization theorem 147
6 Exercises 153
7 Problems 156

Chapter 6. The Gamma and Zeta Functions 159
1 The gamma function 160
1.1 Analytic continuation 161
1.2 Further properties of T 163
2 The zeta function 168
2.1 Functional equation and analytic continuation 168
3 Exercises 174
4 Problems 179

Chapter 7. The Zeta Function and Prime Number Theorem 181
1 Zeros of the zeta function 182
1.1 Estimates for 1/s(s) 187
2 Reduction to the functions v and v1 188
2.1 Proof of the asymptotics for v1 194
Note on interchanging double sums 197
3 Exercises 199
4 Problems 203

Chapter 8. Conformal Mappings 205
1 Conformal equivalence and examples 206
1.1 The disc and upper half-plane 208
1.2 Further examples 209
1.3 The Dirichlet problem in a strip 212
2 The Schwarz lemma; automorphisms of the disc and upper half-plane 218
2.1 Automorphisms of the disc 219
2.2 Automorphisms of the upper half-plane 221
3 The Riemann mapping theorem 224
3.1 Necessary conditions and statement of the theorem 224
3.2 Montel's theorem 225
3.3 Proof of the Riemann mapping theorem 228
4 Conformal mappings onto polygons 231
4.1 Some examples 231
4.2 The Schwarz-Christoffel integral 235
4.3 Boundary behavior 238
4.4 The mapping formula 241
4.5 Return to elliptic integrals 245
5 Exercises 248
6 Problems 254

Chapter 9. An Introduction to Elliptic Functions 261
1 Elliptic functions 262
1.1 Liouville's theorems 264
1.2 The Weierstrass p function 266
2 The modular character of elliptic functions and Eisenstein series 273
2.1 Eisenstein series 273
2.2 Eisenstein series and divisor functions 276
3 Exercises 278
4 Problems 281

Chapter 10. Applications of Theta Functions 283
1 Product formula for the Jacobi theta function 284
1.1 Further transformation laws 289
2 Generating functions 293
3 The theorems about sums of squares 296
3.1 The two-squares theorem 297
3.2 The four-squares theorem 304
4 Exercises 309
5 Problems 314

Appendix A: Asymptotics 318
1 Bessel functions 319
2 Laplace's method; Stirling's formula 323
3 The Airy function 328
4 The partition function 334
5 Problems 341

Appendix B: Simple Connectivity and Jordan Curve Theorem 344
1 Equivalent formulations of simple connectivity 345
2 The Jordan curve theorem 351
2.1 Proof of a general form of Cauchy's theorem 361

Notes and References 365
Bibliography 369
Symbol Glossary 373
Index 375

Elias M. Stein & Rami Shakarchi​ 

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