Title: Theory of Functions of a Complex variable, Volume One
Copyright
1950, BY VERLAG BIRKHAUSER AG., BASEL
1954, BY CHELSEA PUBLISHING COMPANY
TRANSLATOR'S PREFACE
EDITOR'S PREFACE
AUTHOR'S PREFACE
CONTENTS
PART ONE: THE COMPLEX NUMBERS FROM THE ALGEBRAIC POINT OF VIEW
CHAPTER ONE: THE COMPLEX NUMBERS FROM THE ALGEBRAIC POINT OF VIEW
The Discovery of the Complex Numbers (§ 1 )
Definition of the Complex Numbers (§§ 2-9)
Complex Conjugates (§ 10)
Absolute Values (§§ 11-.I2)
Unimodular Numbers (§§ 13-14)
The Amplitude of a Complex Number (§§ 15.17)
Roots (§§ 18-19)
CHAPTER TWO: THE GEOMETRY OF THE COMPLEX NUMBERS
The Gaussian or Complex Plane (§§ 20-22)
Circles in the Complex Plane (§ 23)
The Group of Moebius Transformations (§§ 24-25 )
Circle-Preserving Mappings (§§ 26-28)
Isogonality (§ 29)
The Number Infinity (§ 30)
The Riemann Sphere (§§ 31-33)
Cross·Ratios (§§ 34-37)
Reflection in a Circle (§§ 38-40)
Determination of the Location and Size of a Circle (§§ 41-44)
Pencils of Circles (§§ 45-50)
Moebius Transformations Generated by Two Reflections (§ 51 )
Representation of the General Moebius Transformation as a Product of Inversions in Circles (§§ 52·55 )
CHAPTER THREE: EUCUDEAN, SPHERICAL, AND NON-EUCUDEAN GEOMETRY
Bundles of Circles (§§ 56-57 )
The Equations of the Circles of a Bundle (§§ 58-59)
Products of Inversions in the Circles of a Bundle (§ 60)
The Rigid Motions of Euclidean, Spherical, and Non-Euclidean Geometry (§§ 61-62)
Distance Invariants (§§ 63-65)
Spherical Trigonometry (§§ 66-72)
Non·Euclidean Trigonometry (§§ 73-75)
Spherical Geometry (§ 76)
Elliptic Geometry (§ 77)
The Rotations of the Sphere (§§ 78-79)
Non-Euclidean Geometry (§§ 80-81 )
Non·Euclidean Motions (§§ 82-83 )
Poincare's Half·Plane (§§ 84-85 )
Chordal and Pseudo-Chordal Distance (§§ 86-88)
PART TWO: SOME RESULTS FROM POINT SET THEORY AND FROM TOPOLOGY
CHAPTER ONE: CONVERGENT SEQUENCES OF NUMBERS AND CONTINUOUS COMPLEX FUNCTIONS
Definition of Convergence (§§ 89-90)
Compact Sets of Points (§ 91)
The Cantor Diagonal Process (§ 92 )
Classification of Sets of Points (§§ 93-94 )
Complex Functions (§§ 95-98)
The Boundary Values of a Complex Function (§ 99)
CHAPTER TWO: CURVES AND REGIONS
Connected Sets of Points (§§ 100-101)
Curves (§ 102)
Regions (§ 103)
Neighborhood.Preserving Mappings (§,§ 104-105)
Jordan Curves (§§ 106.109)
Simply and Multiply Connected Regions (§§ 110-113)
CHAPTER THREE: CONTOUR INTEGRATION
Rectifiable Curves (§ 114)
Complex Contour Integrals (§§ 115-119)
The Main Propertiet< of Contour Integrals (§§ 120-122)
The Mean-Value Theorem (§ 123)
PART THREE; ANALYTIC FUNCTIONS
CHAPTER ONE: FOUNDATIONS OF THE THEORY
The Derivative of a Complex Function (§ 124)
Integrable Functions (§§ 125-127)
Definition of Regular Analytic Functions (§ 128)
Cauchy's Theorem (§ 129)
Cauchy's Integral Formula (§§ 130-131)
Some Basic Properties of Analytic Functions (§ 132)
Riemann's Theorem (§§ 133-134)
CHAPTER TWO: THE MAXIMUM·MODULUS PRINCIPLE
The Mean Value of a Function on a Circle (§ 135)
The Maximum-Modulus Principle (§§ 136-139)
Schwarz's Lemma (§§ 140-141)
The Zeros of Regular Analytic Functions (§§ 142-143)
Preservation of Neighborhoods (§ 144)
The Derivative of a Non·Constant Analytic Function Cannot Vanish Identically (§§ 145-146)
CHAPTER THREE: THE POISSON INTEGRAL AND HARMONIC FUNCTIONS
Determination of an Analytic Function by its Real Part (§ 147)
Transformations of Cauchy's Integral for the Circle (§§ 148-149)
Poisson's Integral (§§.150-152)
The Cauchy.Riemann Equations and Harmonic Functions (§§ 153-156)
Harnack's Theorem (§ 157)
Harmonic Measure (§ 158)
An Inequality of Riemann (§ 159)
CHAPTER FOUR: MEROMORPHIC FUNCTIONS
Extension of the Definition of Analytic Functions (§§ 160-161)
Operations with Meromorphic Functions (§§ 162.163)
Partial Fraction Decomposition (§ 164)
Isolated Essential Singularities (§§ 165.166)
Liouville's Theorem and its Application to Polynomials (§§ 167.169)
The Fundamental Theorem of Algebra (§ 170)
Further Properties of Polynomials (§§ 171-173)
PART FOUR: ANALYTIC FUNCTIONS DEFINED BY LIMITING PROCESSES
CHAPTER ONE: CONTINUOUS CONVERGENCE
Continuous Convergence (§§ 174-175)
The Limiting Oscillation (§§ 176-178)
The Normal Kernel of a Sequence of Functions (§ 179)
Comparison of Continuous Convergence with Uniform Convergence (§ 180)
CHAPTER TWO: NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS
The Limiting Oscillation for Sequences ofMeromorphic Functions (§ 181)
Normal Families of Meromorphic Functions (§§ 182-183)
Compact Normal Families (§ 184)
Families of Analytic Functions Uniformly Boundedin the Small (§§ 185-186)
The Limit Functions of Normal Families of Meromorphic Functions (§§ 187-190)
Vitali's Theorem (§ 191)
Uniform Convergence (§ 192)
Osgood's Theorem (§§ 193-194)
Normal Families of Moebius Transformations (§§ 195-197)
The Theorem of A. Hurwitz (§ 198)
A Criterion for Normal Families Bounded in the Small (§ 199)
Simple Functions (§ 200)
CHAPTER THREE: POWER SERIES
Absolutely Convergent Series (§§ 201-204 )
Power Series (§ 205)
The Radius of Convergence (§§ 206-207)
The Taylor Series (§§ 208-209)
Normal Sequences of Power Series (§§ 210-212)
Operations with Power Series (§§ 213-214)
Abel's Transformation (§ 215)
CHAPTER FOUR: PARTIAL-FRACTION DECOMPOSITION AND THE CALCULUS OF RESIDUES
The Laurent Expansion (§§ 216-218)
Analytic Functions with Finitely Many Isolated Singularities (§ 219)
Mittag·Leftler'8 Theorem (§§ 220-222)
Meromorphic Functions with Prescribed Simple1 Poles (§ 223 )
The Residue and its Applications (§§ 224-225)
The Number of Zeros of a Function, and Rouche's Theorem (§ 226)
The Inverse of an Analytic Function (§§ 227-228)
Lagrange's Series (§§ 229-230)
Kepler's Equation (§ 231)
The Monodromy Theorem (§§ 232-233)
PART FIVE: SPECIAL FUNCTIONS
CHAPTER ONE: THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
The Exponential Function e^z (§ 234)
The Trigonometric Functions (§§ 235.237)
The 'Periods of the Exponential Functions (§§ 238-239)
The Hyperbolic Functions (§ 240)
Periods and Fundamental Regions of the Trigonometric Functions (§§ 241-242)
The Functions tg z and tgh z (§§ 243-244)
Numerical Calculation of :pi (§ 245 )
CHAPTER TWO: THE LOGARITHMIC FUNCTION AND THE GENERAL POWER FUNCTION
The Natural Logarithm (§§ 246-250)
Series Expansions and Numerical Calculation of the Logarithm (§§ 251-253)
The General Power Function (§§ 254-255)
Regular Functions with a Many.Valued Inverse (§ 256 )
Bounds for n! (§ 257)
Bounds for the Series (§ 258)
The Partial.Fraction Decomposition of pi ctg (pi z) (§§ 259-261)
The Product Formula for sin (pi z) and Wallis' Formula (§ 262)
CHAPTER THREE: TIlE BERNOULLI NUMBERS AND TIlE GAMMA FUNCTION
The Inverse of Differencing (§ 263)
The Bernoulli Numbers (§ 264)
The Symbolic Calculus of E. Lucas (§§ 265.268)
Clausen'8 Theorem (§ 269)
Euler's Constant (§ 270)
The Function Gamma (z) (§§ 271.273)
The Bohr.Mollerup Theorem (§§ 274·275 )
Stirling's Series (§§ 276.277)
Gauss's Product Formula (§ 278)
Compilation of Formulas; Applications (§§ 279.280)
INDEX
CHELSEA SCIENTIFIC BOOKS

lgrsnf/Caratheodory C. Theory of functions of a complex variable. Vol.1 (Chelsea, 1954)(ASIN B000JL925E)(600dpi)(K)(T)(O)(319s)_MCc_.djvu

2024-07-27