Theory of Functions of a Complex variable, Volume One

Title: Theory of Functions of a Complex variable, Volume One......Page 1
1954, BY CHELSEA PUBLISHING COMPANY......Page 2
TRANSLATOR\'S PREFACE......Page 3
EDITOR\'S PREFACE......Page 5
AUTHOR\'S PREFACE......Page 7
CONTENTS......Page 9
PART ONE: THE COMPLEX NUMBERS FROM THE ALGEBRAIC POINT OF VIEW......Page 13
The Discovery of the Complex Numbers (§ 1 )......Page 14
Definition of the Complex Numbers (§§ 2-9)......Page 15
Complex Conjugates (§ 10)......Page 21
Absolute Values (§§ 11-.I2)......Page 22
Unimodular Numbers (§§ 13-14)......Page 23
The Amplitude of a Complex Number (§§ 15.17)......Page 26
Roots (§§ 18-19)......Page 27
The Gaussian or Complex Plane (§§ 20-22)......Page 30
Circles in the Complex Plane (§ 23)......Page 33
The Group of Moebius Transformations (§§ 24-25 )......Page 34
Circle-Preserving Mappings (§§ 26-28)......Page 36
The Number Infinity (§ 30)......Page 38
The Riemann Sphere (§§ 31-33)......Page 40
Cross·Ratios (§§ 34-37)......Page 42
Reflection in a Circle (§§ 38-40)......Page 45
Determination of the Location and Size of a Circle (§§ 41-44)......Page 48
Pencils of Circles (§§ 45-50)......Page 51
Moebius Transformations Generated by Two Reflections (§ 51 )......Page 54
Representation of the General Moebius Transformation as a Product of Inversions in Circles (§§ 52·55 )......Page 56
Bundles of Circles (§§ 56-57 )......Page 59
The Equations of the Circles of a Bundle (§§ 58-59)......Page 60
Products of Inversions in the Circles of a Bundle (§ 60)......Page 62
The Rigid Motions of Euclidean, Spherical, and Non-Euclidean Geometry (§§ 61-62)......Page 63
Distance Invariants (§§ 63-65)......Page 65
Spherical Trigonometry (§§ 66-72)......Page 68
Non·Euclidean Trigonometry (§§ 73-75)......Page 75
Spherical Geometry (§ 76)......Page 79
Elliptic Geometry (§ 77)......Page 82
The Rotations of the Sphere (§§ 78-79)......Page 84
Non-Euclidean Geometry (§§ 80-81 )......Page 86
Non·Euclidean Motions (§§ 82-83 )......Page 89
Poincare\'s Half·Plane (§§ 84-85 )......Page 92
Chordal and Pseudo-Chordal Distance (§§ 86-88)......Page 94
PART TWO: SOME RESULTS FROM POINT SET THEORY AND FROM TOPOLOGY......Page 99
Definition of Convergence (§§ 89-90)......Page 100
Compact Sets of Points (§ 91)......Page 102
The Cantor Diagonal Process (§ 92 )......Page 103
Classification of Sets of Points (§§ 93-94 )......Page 104
Complex Functions (§§ 95-98)......Page 105
The Boundary Values of a Complex Function (§ 99)......Page 108
Connected Sets of Points (§§ 100-101)......Page 109
Curves (§ 102)......Page 110
Regions (§ 103)......Page 111
Neighborhood.Preserving Mappings (§,§ 104-105)......Page 112
Jordan Curves (§§ 106.109)......Page 113
Simply and Multiply Connected Regions (§§ 110-113)......Page 115
Rectifiable Curves (§ 114)......Page 119
Complex Contour Integrals (§§ 115-119)......Page 120
The Main Propertiet< of Contour Integrals (§§ 120-122)......Page 126
The Mean-Value Theorem (§ 123)......Page 127
PART THREE; ANALYTIC FUNCTIONS......Page 131
The Derivative of a Complex Function (§ 124)......Page 132
Integrable Functions (§§ 125-127)......Page 133
Definition of Regular Analytic Functions (§ 128)......Page 137
Cauchy\'s Theorem (§ 129)......Page 138
Cauchy\'s Integral Formula (§§ 130-131)......Page 140
Riemann\'s Theorem (§§ 133-134)......Page 143
The Mean Value of a Function on a Circle (§ 135)......Page 145
The Maximum-Modulus Principle (§§ 136-139)......Page 146
Schwarz\'s Lemma (§§ 140-141)......Page 148
The Zeros of Regular Analytic Functions (§§ 142-143)......Page 150
Preservation of Neighborhoods (§ 144)......Page 152
The Derivative of a Non·Constant Analytic Function Cannot Vanish Identically (§§ 145-146)......Page 153
Determination of an Analytic Function by its Real Part (§ 147)......Page 155
Transformations of Cauchy\'s Integral for the Circle (§§ 148-149)......Page 156
Poisson\'s Integral (§§.150-152)......Page 159
The Cauchy.Riemann Equations and Harmonic Functions (§§ 153-156)......Page 162
Harnack\'s Theorem (§ 157)......Page 166
Harmonic Measure (§ 158)......Page 167
An Inequality of Riemann (§ 159)......Page 170
Extension of the Definition of Analytic Functions (§§ 160-161)......Page 171
Operations with Meromorphic Functions (§§ 162.163)......Page 172
Partial Fraction Decomposition (§ 164)......Page 174
Isolated Essential Singularities (§§ 165.166)......Page 175
Liouville\'s Theorem and its Application to Polynomials (§§ 167.169)......Page 177
The Fundamental Theorem of Algebra (§ 170)......Page 180
Further Properties of Polynomials (§§ 171-173)......Page 181
PART FOUR: ANALYTIC FUNCTIONS DEFINED BY LIMITING PROCESSES......Page 185
Continuous Convergence (§§ 174-175)......Page 186
The Limiting Oscillation (§§ 176-178)......Page 188
Comparison of Continuous Convergence with Uniform Convergence (§ 180)......Page 191
The Limiting Oscillation for Sequences ofMeromorphic Functions (§ 181)......Page 193
Normal Families of Meromorphic Functions (§§ 182-183)......Page 195
Compact Normal Families (§ 184)......Page 196
Families of Analytic Functions Uniformly Boundedin the Small (§§ 185-186)......Page 197
The Limit Functions of Normal Families of Meromorphic Functions (§§ 187-190)......Page 198
Uniform Convergence (§ 192)......Page 202
Osgood\'s Theorem (§§ 193-194)......Page 203
Normal Families of Moebius Transformations (§§ 195-197)......Page 205
The Theorem of A. Hurwitz (§ 198)......Page 207
A Criterion for Normal Families Bounded in the Small (§ 199)......Page 208
Simple Functions (§ 200)......Page 209
Absolutely Convergent Series (§§ 201-204 )......Page 211
Power Series (§ 205)......Page 214
The Radius of Convergence (§§ 206-207)......Page 215
The Taylor Series (§§ 208-209)......Page 218
Normal Sequences of Power Series (§§ 210-212)......Page 220
Operations with Power Series (§§ 213-214)......Page 224
Abel\'s Transformation (§ 215)......Page 227
The Laurent Expansion (§§ 216-218)......Page 230
Analytic Functions with Finitely Many Isolated Singularities (§ 219)......Page 233
Mittag·Leftler\'8 Theorem (§§ 220-222)......Page 235
Meromorphic Functions with Prescribed Simple1 Poles (§ 223 )......Page 238
The Residue and its Applications (§§ 224-225)......Page 239
The Number of Zeros of a Function, and Rouche\'s Theorem (§ 226)......Page 241
The Inverse of an Analytic Function (§§ 227-228)......Page 242
Lagrange\'s Series (§§ 229-230)......Page 244
Kepler\'s Equation (§ 231)......Page 248
The Monodromy Theorem (§§ 232-233)......Page 249
PART FIVE: SPECIAL FUNCTIONS......Page 253
The Exponential Function e^z (§ 234)......Page 254
The Trigonometric Functions (§§ 235.237)......Page 256
The \'Periods of the Exponential Functions (§§ 238-239)......Page 260
The Hyperbolic Functions (§ 240)......Page 261
Periods and Fundamental Regions of the Trigonometric Functions (§§ 241-242)......Page 263
The Functions tg z and tgh z (§§ 243-244)......Page 265
Numerical Calculation of :\\pi (§ 245 )......Page 267
The Natural Logarithm (§§ 246-250)......Page 269
Series Expansions and Numerical Calculation of the Logarithm (§§ 251-253)......Page 273
The General Power Function (§§ 254-255)......Page 276
Regular Functions with a Many.Valued Inverse (§ 256 )......Page 278
Bounds for n! (§ 257)......Page 279
Bounds for the Series (§ 258)......Page 280
The Partial.Fraction Decomposition of  \\pi ctg (\\pi z)  (§§ 259-261)......Page 281
The Product Formula for sin (\\pi z) and Wallis\' Formula (§ 262)......Page 284
The Inverse of Differencing (§ 263)......Page 286
The Bernoulli Numbers (§ 264)......Page 288
The Symbolic Calculus of E. Lucas (§§ 265.268)......Page 289
Clausen\'8 Theorem (§ 269)......Page 294
Euler\'s Constant (§ 270)......Page 297
The Function \\Gamma (z) (§§ 271.273)......Page 299
The Bohr.Mollerup Theorem (§§ 274·275 )......Page 301
Stirling\'s Series (§§ 276.277)......Page 304
Gauss\'s Product Formula (§ 278)......Page 308
Compilation of Formulas; Applications (§§ 279.280)......Page 309
INDEX......Page 312
CHELSEA SCIENTIFIC BOOKS......Page 315