**Graduate Studies in Mathematics**

Volume: 202;
2019;
480 pp;
Hardcover

MSC: Primary 30;

**Print ISBN: 978-1-4704-5286-5
Product Code: GSM/202**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

**Electronic ISBN: 978-1-4704-5448-7
Product Code: GSM/202.E**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

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#### Supplemental Materials

# Introduction to Complex Analysis

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*Michael E. Taylor*

In this text, the reader will learn that all the basic functions that
arise in calculus—such as powers and fractional powers, exponentials
and logs, trigonometric functions and their inverses, as well as many
new functions that the reader will meet—are naturally defined for
complex arguments. Furthermore, this expanded setting leads to a much
richer understanding of such functions than one could glean by merely
considering them in the real domain. For example, understanding the
exponential function in the complex domain via its differential
equation provides a clean path to Euler's formula and hence to a
self-contained treatment of the trigonometric functions. Complex
analysis, developed in partnership with Fourier analysis, differential
equations, and geometrical techniques, leads to the development of a
cornucopia of functions of use in number theory, wave motion,
conformal mapping, and other mathematical phenomena, which the reader
can learn about from material presented here.

This book could serve for either a one-semester course or a
two-semester course in complex analysis for beginning graduate
students or for well-prepared undergraduates whose background includes
multivariable calculus, linear algebra, and advanced calculus.

#### Readership

Graduate students interested in complex analysis.

#### Table of Contents

# Table of Contents

## Introduction to Complex Analysis

- Cover Cover11
- Title page i2
- Preface vii8
- Some basic notation xiii14
- Chapter 1. Basic calculus in the complex domain 116
- 1.1. Complex numbers, power series, and exponentials 318
- Exercises 1227
- 1.2. Holomorphic functions, derivatives, and path integrals 1429
- Exercises 2439
- 1.3. Holomorphic functions defined by power series 2641
- Exercises 3247
- 1.4. Exponential and trigonometric functions: Euler’s formula 3449
- Exercises 4156
- 1.5. Square roots, logs, and other inverse functions 4459
- Exercises 4863
- 1.6. Pi is irrational 5873

- Chapter 2. Going deeper –the Cauchy integral theorem and consequences 6176
- 2.1. The Cauchy integral theorem and the Cauchy integral formula 6378
- Exercises 7186
- 2.2. The maximum principle, Liouville’s theorem, and the fundamental theorem of algebra 7388
- Exercises 7691
- 2.3. Harmonic functions on planar domains 7893
- Exercises 87102
- 2.4. Morera’s theorem, the Schwarz reflection principle, and Goursat’s theorem 89104
- Exercises 92107
- 2.5. Infinite products 93108
- Exercises 106121
- 2.6. Uniqueness and analytic continuation 107122
- Exercises 112127
- 2.7. Singularities 114129
- Exercises 117132
- 2.8. Laurent series 118133
- Exercises 122137
- 2.9. Green’s theorem 123138
- 2.10. The fundamental theorem of algebra (elementary proof) 128143
- 2.11. Absolutely convergent series 130145

- Chapter 3. Fourier analysis and complex function theory 135150
- 3.1. Fourier series and the Poisson integral 137152
- Exercises 150165
- 3.2. Fourier transforms 152167
- Exercises 159174
- More general sufficient condition for 𝑓∈\Cal{𝐴}(\RR) 160175
- Fourier uniqueness 162177
- 3.3. Laplace transforms and Mellin transforms 163178
- Exercises 165180
- The matrix Laplace transform and Duhamel’s formula 167182
- 3.4. Inner product spaces 169184
- 3.5. The matrix exponential 172187
- 3.6. The Weierstrass and Runge approximation theorems 174189

- Chapter 4. Residue calculus, the argument principle, and two very special functions 183198
- 4.1. Residue calculus 186201
- Exercises 193208
- 4.2. The argument principle 196211
- Exercises 201216
- 4.3. The Gamma function 203218
- Exercises 207222
- The Legendre duplication formula 209224
- 4.4. The Riemann zeta function and the prime number theorem 211226
- Counting primes 219234
- The prime number theorem 221236
- Exercises 225240
- 4.5. Euler’s constant 227242
- 4.6. Hadamard’s factorization theorem 233248

- Chapter 5. Conformal maps and geometrical aspects of complex function theory 243258
- 5.1. Conformal maps 246261
- Exercises 254269
- 5.2. Normal families 255270
- Exercises 256271
- 5.3. The Riemann sphere and other Riemann surfaces 257272
- Exercises 265280
- 5.4. The Riemann mapping theorem 268283
- Exercises 271286
- 5.5. Boundary behavior of conformal maps 272287
- Exercises 275290
- 5.6. Covering maps 277292
- Exercises 279294
- 5.7. The disk covers the twice-punctured plane 280295
- Exercises 281296
- 5.8. Montel’s theorem 284299
- Exercises on Fatou sets and Julia sets\nopunct 286301
- 5.9. Picard’s theorem 289304
- Exercises 290305
- 5.10. Harmonic functions II 290305
- Exercises 302317
- 5.11. Surfaces and metric tensors 303318
- 5.12. Poincaré metrics 311326
- 5.13. Groups 318333

- Chapter 6. Elliptic functions and elliptic integrals 323338
- 6.1. Periodic and doubly periodic functions 325340
- Exercises 328343
- 6.2. The Weierstrass P-function in elliptic function theory 331346
- Exercises 334349
- 6.3. Theta functions and elliptic functions 337352
- Exercises 341356
- 6.4. Elliptic integrals 342357
- Exercises 348363
- 6.5. The Riemann surface of the square root of a cubic 350365
- Exercises 355370
- 6.6. Rapid evaluation of the Weierstrass P-function 358373
- Rectangular lattices 359374

- Chapter 7. Complex analysis and differential equations 363378
- Appendix A. Complementary material 417432
- A.1. Metric spaces, convergence, and compactness 418433
- Exercises 430445
- A.2. Derivatives and diffeomorphisms 431446
- A.3. The Laplace asymptotic method and Stirling’s formula 437452
- A.4. The Stieltjes integral 441456
- A.5. Abelian theorems and Tauberian theorems 448463
- A.6. Cubics, quartics, and quintics 459474

- Bibliography 473488
- Index 477492
- Back Cover Back Cover1497