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1. Complex analysis [2003]
 Stein, Elias M., 1931
 Princeton, N.J. ; Woodstock : Princeton University Press, 2003.
 Description
 Book — xvii, 379 p. : ill. ; 23cm.
 Summary

 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 PaleyWiener 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 halfplane
 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 halfplane
 218 2.1 Automorphisms of the disc
 219 2.2 Automorphisms of the upper halfplane
 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 SchwarzChristoffel 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 twosquares theorem
 297 3.2 The foursquares 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.
 (source: Nielsen Book Data)
(source: Nielsen Book Data) 9780691113852 20160528
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Science Library (Li and Ma)
Science Library (Li and Ma)  Status 

Stacks  
QA331.7 .S74 2003  Unknown On reserve at Li and Ma Science Library 2hour loan 
MATH11601
 Course
 MATH11601  Complex Analysis
 Instructor(s)
 Eliashberg, Yakov