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1. Complex analysis [2003]

Book
xvii, 379 p. : ill. ; 23cm.
  • 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.
  • (source: Nielsen Book Data)9780691113852 20160528
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.
(source: Nielsen Book Data)9780691113852 20160528
Science Library (Li and Ma)
MATH-116-01
Book
xiii, 317 p. : ill. ; 25 cm.
Presents a basic introduction to complex analysis.
(source: Nielsen Book Data)9780387903286 20160528
Science Library (Li and Ma)
MATH-116-01

3. Complex variables [1990]

Book
xiv, 427 p. : ill. ; 25 cm.
Science Library (Li and Ma)
MATH-116-01
Book
xiv, 331 p. : ill. ; 24 cm.
  • Chapter 1: Complex Numbers1 The Algebra of Complex Numbers1.1 Arithmetic Operations1.2 Square Roots1.3 Justification1.4 Conjugation, Absolute Value1.5 Inequalities2 The Geometric Representation of Complex Numbers2.1 Geometric Addition and Multiplication2.2 The Binomial Equation2.3 Analytic Geometry2.4 The Spherical RepresentationChapter 2: Complex Functions1 Introduction to the Concept of Analytic Function1.1 Limits and Continuity1.2 Analytic Functions1.3 Polynomials1.4 Rational Functions2 Elementary Theory of Power Series2.1 Sequences2.2 Series2.3 Uniform Coverages2.4 Power Series2.5 Abel's Limit Theorem3 The Exponential and Trigonometric Functions3.1 The Exponential3.2 The Trigonometric Functions3.3 The Periodicity3.4 The LogarithmChapter 3: Analytic Functions as Mappings1 Elementary Point Set Topology1.1 Sets and Elements1.2 Metric Spaces1.3 Connectedness1.4 Compactness1.5 Continuous Functions1.6 Topological Spaces2 Conformality2.1 Arcs and Closed Curves2.2 Analytic Functions in Regions2.3 Conformal Mapping2.4 Length and Area3 Linear Transformations3.1 The Linear Group3.2 The Cross Ratio3.3 Symmetry3.4 Oriented Circles3.5 Families of Circles4 Elementary Conformal Mappings4.1 The Use of Level Curves4.2 A Survey of Elementary Mappings4.3 Elementary Riemann SurfacesChapter 4: Complex Integration1 Fundamental Theorems1.1 Line Integrals1.2 Rectifiable Arcs1.3 Line Integrals as Functions of Arcs1.4 Cauchy's Theorem for a Rectangle1.5 Cauchy's Theorem in a Disk2 Cauchy's Integral Formula2.1 The Index of a Point with Respect to a Closed Curve2.2 The Integral Formula2.3 Higher Derivatives3 Local Properties of Analytical Functions3.1 Removable Singularities. Taylor's Theorem3.2 Zeros and Poles3.3 The Local Mapping3.4 The Maximum Principle4 The General Form of Cauchy's Theorem4.1 Chains and Cycles4.2 Simple Connectivity4.3 Homology4.4 The General Statement of Cauchy's Theorem4.5 Proof of Cauchy's Theorem4.6 Locally Exact Differentials4.7 Multiply Connected Regions5 The Calculus of Residues5.1 The Residue Theorem5.2 The Argument Principle5.3 Evaluation of Definite Integrals6 Harmonic Functions6.1 Definition and Basic Properties6.2 The Mean-value Property6.3 Poisson's Formula6.4 Schwarz's Theorem6.5 The Reflection PrincipleChapter 5: Series and Product Developments1 Power Series Expansions1.1 Wierstrass's Theorem1.2 The Taylor Series1.3 The Laurent Series2 Partial Fractions and Factorization2.1 Partial Fractions2.2 Infinite Products2.3 Canonical Products2.4 The Gamma Function2.5 Stirling's Formula3 Entire Functions3.1 Jensen's Formula3.2 Hadamard's Theorem4 The Riemann Zeta Function4.1 The Product Development4.2 Extension of '(s) to the Whole Plane4.3 The Functional Equation4.4 The Zeros of the Zeta Function5 Normal Families5.1 Equicontinuity5.2 Normality and Compactness5.3 Arzela's Theorem5.4 Families of Analytic Functions5.5 The Classical DefinitionChapter 6: Conformal Mapping, Dirichlet's Problem1 The Riemann Mapping Theorem1.1 Statement and Proof1.2 Boundary Behavior1.3 Use of the Reflection Principle1.4 Analytic Arcs2 Conformal Mapping of Polygons2.1 The Behavior at an Angle2.2 The Schwarz-Christoffel Formula2.3 Mapping on a Rectangle2.4 The Triangle Functions of Schwarz3 A Closer Look at Harmonic Functions3.1 Functions with Mean-value Property3.2 Harnack's Principle4 The Dirichlet Problem4.1 Subharmonic Functions4.2 Solution of Dirichlet's Problem5 Canonical Mappings of Multiply Connected Regions5.1 Harmonic Measures5.2 Green's Function5.3 Parallel Slit RegionsChapter 7: Elliptic Functions1 Simply Periodic Functions1.1 Representation by Exponentials1.2 The Fourier Development1.3 Functions of Finite Order2 Doubly Periodic Functions2.1 The Period Module2.2 Unimodular Transformations2.3 The Canonical Basis2.4 General Properties of Elliptic Functions3 The Weierstrass Theory3.1 The Weierstrass p-function3.2 The Functions '(z) and s(z)3.3 The Differential Equation3.4 The Modular Function '(r)3.5 The Conformal Mapping by '(r)Chapter 8: Global Analytic Functions1 Analytic Continuation1.1 The Weierstrass Theory1.2 Germs and Sheaves1.3 Sections and Riemann Surfaces1.4 Analytic Continuations along Arcs1.5 Homotopic Curves1.6 The Monodromy Theorem1.7 Branch Points2 Algebraic Functions2.1 The Resultant of Two Polynomials2.2 Definition and Properties of Algebraic Functions2.3 Behavior at the Critical Points3 Picard's Theorem3.1 Lacunary Values4 Linear Differential Equations4.1 Ordinary Points4.2 Regular Singular Points4.3 Solutions at Infinity4.4 The Hypergeometric Differential Equation4.5 Riemann's Point of ViewIndex.
  • (source: Nielsen Book Data)9780070006577 20160528
A standard source of information of functions of one complex variable, this text has retained its wide popularity in this field by being consistently rigorous without becoming needlessly concerned with advanced or overspecialized material. Difficult points have been clarified, the book has been reviewed for accuracy, and notations and terminology have been modernized. Chapter 2, Complex Functions, features a brief section on the change of length and area under conformal mapping, and much of Chapter 8, Global-Analytic Functions, has been rewritten in order to introduce readers to the terminology of germs and sheaves while still emphasizing that classical concepts are the backbone of the theory. Chapter 4, Complex Integration, now includes a new and simpler proof of the general form of Cauchy's theorem. There is a short section on the Riemann zeta function, showing the use of residues in a more exciting situation than in the computation of definite integrals.
(source: Nielsen Book Data)9780070006577 20160528
Science Library (Li and Ma)
MATH-116-01