Theory of Functions of a Complex Variable, Part 11

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American Mathematical Soc., 2005 - Mathematics - 1138 pages
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Theorems are presented in a logical way and are carefully proved, making this a most useful book for students. --Choice This magnificent textbook, translated from the Russian, was first published in 1965-1967. The book covers all aspects of the theory of functions of one complex variable. The chosen proofs give the student the best `feel' for the subject. The watchwords are clarity and straightforwardness. The author was a leading Soviet function-theorist: It is seldom that an expert of his stature puts himself so wholly at the service of the student. This book includes over 150 illustrations and 700 exercises.
  

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Contents

BASIC CONCEPTS
3
1
5
1
7
ELEMENTARY MEROMORPHIC FUNCTIONS
15
THE CALCULUS OF RESIDUES AND
40
The CirclePreserving Property of Mobius Transformations 168
45
Fixed Points of a Mobius Transformation
46
CONNECTEDNESS CURVES AND DOMAINS
59
2
3
12
7
DIFFERENTIATION ELEMENTARY FUNC
11
Connected Sets Continuous Curves and Con
15
8
18
SETS AND FUNCTIONS LIMITS AND CON
23
Convergence of the Real and Imaginary Parts
32
Limit Points of Sets Bounded Sets 42
42

11 Interpolation Theory
67
The Relation between Power Series and Fourier
73
TT O INVERSE AND IMPLICIT FUNCTIONS
86
UNIVALENT FUNCTIONS Page
115
Mapping of the Upper HalfPlane onto
124
21 Sufficient Conditions for Univalent Mapping
135
5
143
2
145
3
153
variance of the Cross Ratio
171
APPLICATIONS TO FLUID DYNAMICS
174
Mapping of a Circle onto a Circle
176
Symmetry Transformations
178
Examples
180
Examples
181
Lobachevskian Geometry
183
7
189
The Mapping w z +
197
Transcendental Meromorphic Functions Trig onometric Functions
202
Probems
207
ELEMENTARY MULTIPLEVALUED FUNC TIONS Page
212
SingleValued Branches Univalent Functions
213
The Mapping w z
214
The Mapping h PUl
219
The Logarithm
224
The Function za Exponentials and Logarithms to an Arbitrary Base
228
The Mapping w Arc cos z
234
Functions of Bounded Characteristic
236
The Mapping w z + In z
237
Problems
239
9
249
Coefficients
255
10
282
Hadamards Factorization Theorem
289
CAUCHYS INTEGRAL AND RELATED TOP
293
Meromorphic Functions
297
The Gamma Function
304
Boundary Values of Integrals of the Cauchy
306
53 Integral Representations of Tr Partial
310
16
321
Simply and Multiply Connected Domains 66
66
Some Further Results 72
72
Stereographic Projection Sets of Points on
79
APPROXIMATION BY RATIONAL FUNC
80
Expansion of an Analytic Function in Power
81
Conformality of Stereographic Projection Con
87
Runges Theorem and Related Results 88
88
HOMEOMORPHISMS Page 94
94
Approximation on Closed Domains 97
97
Faber Polynomials 104
104
Bernsteins Theorem 112
112
Approximation in the Mean 116
116
GEOMETRIC INTERPRETATION OF THE
118
Conformal Mapping of the Extended Plane 124
124
The Mapping w Pnz 130
130
5
135
The Mapping w ez 140
140
The Mapping w cos z 150
150
WEIERSTRASS
155
ELEMENTARY MEROMORPHIC FUNCTIONS
160
The Functions pz a iP and pz a t5
162
The Differential Equation for pz 168
168
The Functions z and az 178
178
The Spherical Pendulum 186
186
Rational Functions 160
192
JACOBIS THEORY
194
7
217
RECTIFIABLE CURVES COMPLEX INTE GRALS Page
245
Integrals of Complex Functions
248
Properties of Complex Integrals 250
250
Problems 253
253
ANALYTIC CONTINUATION Page 257
257
Analytic Continuation in a Star 272
272
RIEMANN SURFACES ANALYTIC CON
278
The Analytic Configuration as a Topological
287
THE SYMMETRY PRINCIPLE AND
315
Examples
332
RUDIMENTS Page
344
BIBLIOGRAPHY Page
351
RAMIFICATIONS Page
359

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Complex Analysis
Joseph Bak
Limited preview - 1997
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