## Theory of Functions of a Complex Variable, Part 11Theorems 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 |

351 | |

359 | |

### Common terms and phrases

absolutely convergent accessible points according to Theorem analytic function analytic on G boundary points bounded closed Jordan curve closed rectifiable Jordan compact subset conformal mapping const contained converges uniformly Corollary deleted neighborhood denote Dirichlet series domain G entire function equation essential singular point Example fact Figure final point finite number fn(z formula function f(z given harmonic function hence implies inequality infinite inside integral interval Jordan arc Jordan curve Jordan half-intervals Laurent expansion lemma Let f(z Let G limit point maximum modulus principle meromorphic Moreover obtain obviously Pn(z point of f(z point of G point z0 pole of order poles of f(z prime end Problem Prove real axis rectifiable Jordan curve regular point satisfies the conditions simply connected simply connected domain single-valued sufficiently large suppose f(z uniformly convergent unit disk univalent velocity