Analytic Function Theory, Volume II

Front Cover
Foreword
Contents
Volume I
Volume II
Symbols
10 ANALYTIC CONTINUATION
10.1. Introduction.
10.2. Rearrangements of power series.
10.3. Analytic functions.
10.4. Singularities.
10.5. Borel monogenic functions.
10.6. Multivalued functions and Riemann surfaces.
10.7. Law of permanence of functional equations.
COLLATERAL READING
11 SINGULARITIES AND REPRESENTATION OF ANALYTIC FUNCTIONS
11.1. Holomorphy-preserving transformations: I. Integral operators.
11.2. Holomorphy-preserving transformations: Il. Differential operators.
11.3. Power series with analytic coefficients.
11.4. Analytic continuation in a star.
11.5. Polynomial series.
11.6. Composition theorems.
11.7. Gap theorems and nonoontinuable power series.
COLLATERAL READING
12 ALGEBRAIC FUNCTIONS
12.1. Local properties.
12.2. Critical points.
12.3. Newton'
s diagram.
12.4. Riemann surfaces
some concepts of algebraic geometry.
12.5. Rational functions on the surface and Abelian integrals.
COLLATERAL READING
13 ELLIPTIC FUNCTIONS
13.1. Doubly-periodic functions.
13.2. The functions of Weierstrass.
13.3. Some further properties of elliptic functions.
13.4. On the functions of Jacobi.
13.5. The theta functions.
13.6. Modular functions.
COLLATERAL READING
14 ENTIRE AND MEROMORPHIC FUNCTIONS
14.1. Order relations for entire functions.
14.2. Entire functions of finite order.
14.3. Functions with real zeros.
14.4. Characteristic functions.
14.5. Picard'
s and Landau'
s theorems.
14.6. The second fundamental theorem.
14.7. Defect relations.
COLLATERAL READING
15 NORMAL FAMILIES
15.1. Schwarz'
s lemma and hyperbolic measure.
15.2. Normal families.
15.3. Induced convergence.
15.4. Applications.
COLLATERAL READING
16 LEMNISCATES
16.1. Chebichev polynomials.
16.2. The transfinite diameter.
16.3. Additive set functions
Radon-Stieltjes integrals.
16.4. Logarithmic capacity.
16.5. Green'
s function
Hilbert'
s theorem.
16.6. Runge'
s theorem.
16. 7. Overconvergence.
COLLATERAL READING
17 CONFORMAL MAPPING
17.1. Riemann'
s mapping theorem.
17.2. The kernel function.
17.3. Fekete polynomials and the exterior mapping problem.
17.4. Univalent functions.
17.5. The boundary problem.
17.6. Special mappings.
17.7. The theorem of Bloch.
COLLATERAL READING
18 MAJORIZATION
18.1. The Phragmen-Lindelof Principle.
18.2. Dirichlet'
s problem
Lindelof'
s principle.
18.3. Harmonic measure.
18.4. The Nevanlinna-Ahlfors-Heins theorems.
18.5. Subordination.
COLLATERAL READING
19 FUNCTIONS HOLOMORPHIC IN A HALF-PLANE
19.1. The Hardy-Lebesgue classes.
19.2. Bounded functions.
19.3. Growth-measuring functions.
19.4. Remarks on Laplace-Stieltjes integrals.
COLLATERAL READING
Bibliography
Index
Back Cover