# Generalized Analytic Functions

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Generalized Analytic Functions is concerned with foundations of the general theory of generalized analytic functions and some applications to problems of differential geometry and theory of shells. Some classes of functions and operators are discussed, along with the reduction of a positive differential quadratic form to the canonical form. Boundary value problems and infinitesimal bendings of surfaces are also considered. Comprised of six chapters, this volume begins with a detailed treatment of various problems of the general theory of generalized analytic functions as as well as boundary value problems. The reader is introduced to some classes of functions and functional spaces, with emphasis on functions of two independent variables. Subsequent chapters focus on the problem of reducing a positive differential quadratic form to the canonical form; basic properties of solutions of elliptic systems of partial differential equations of the first order, in a two-dimensional domain; and some boundary value problems for an elliptic system of equations of the first order and for an elliptic equation of the second order, in a two-dimensional domain. The final part of the book deals with problems of the theory of surfaces and the membrane theory of shells. This book is intended for students of advanced courses of the mechanico-mathematical faculties, postgraduates, and research workers.

## Table of Contents

Foreword

Part One Foundations of the General Theory of Generalized Analytic Functions and Boundary Value Problems

Chapter I. Some Classes of Functions and Operators

§1. Classes of Functions and Functional Spaces

§2. Classes of Curves and Domains. Some Properties of Conformal Mapping

§3. Some Properties of the Cauchy Type Integral

§4. Non-Homogeneous Cauchy-Riemann System

§5. Generalized Derivatives in the Sobolev Sense and their Properties

§6. Properties of the Operator TGf

§7. Green Identity for the Class of Functions D1,p. Areal Derivative

§8. On Differential Properties of Functions of the Form TGf. Operator IIf

§9. Extension of the Operator IIf

§10. Some Other Properties of Functions of the Classes DZ(G) and B-z(G)

Chapter II. Reduction of a Positive Differential Quadratic Form to the Canonical Form. Beltrami's Equation. Geometric Applications

§1. Introductory Remarks. Homeomorphisms of a Quadratic Form

§2. Beltrami's System of Equations

§3. Construction of the Basic Homeomorphism of Beltrami's Equation

§4. Proof of Existence of a Local Homeomorphism

§5. Proof of Existence of a Complete Homeomorphism

§6. Reduction of a Positive Quadratic Differential Form to the Canonical Form. Isometric and Isometric-Conjugate Corodinate Systems on a Surface

§7. Reduction of Equations of the Elliptic Type to the Canonical Form

Chapter III. Foundations of the General Theory of Generalized Analytic Functions

§1. Basic Concepts, Terminology and Notations

§2. Integral Equation for Functions of the Class U(A, B, F, G)

§3. Continuity and Differential Properties of Functions of the Class Up(G)

§4. Basic Lemma. Generalizations of some Classical Theorems

§5. Integral Representation of the Second Kind for Generalized Analytic Functions

§6. Generating Pair of Functions of the Class Up,2(A, B, E) Derivative in the Bers Sense

§7. Inversion of the Non-Linear Integral Equation (4.3)

§8. System of Fundamental Generalized Analytic Functions and Fundamental Kernels of the Class Up,2(A, B, G), p > 2

§9. Adjoint Equation. Green's Identity. Equations of the Second Order

§10. Generalized Cauchy Formula

§11. Continuous Continuations of Generalized Analytic Functions. Generalized Principle of Symmetry

§12. Compactness

§13. Representation of Resolvent by Means of Kernels

§14. Representation of Generalized Analytic Functions by Means of Generalized Integrals of the Cauchy Type

§15. Complete Systems of Generalized Analytic Functions. Generalized Power Series

§16. Integral Equations for the Real Part of a Generalized Analytic Function

§17. Properties of Solutions of Elliptic System of Equations of the General Form

Chapter IV. Boundary Value Problems

§1. Formulation of the Generalized Riemann-Hilbert Problem. Continuity Properties of the Solution of the Problem

§2. The Adjoint Boundary Value Problem Å. Necessary and Sufficient Conditions of Solubility of Problem A

§3. Index of Problem A. Keduction of the Boundary Condition of Problem A to the Canonical Form

§4. Properties of the Zeros of Solutions of the Homogeneous Problem Å. Criteria of Solubility of the Problems Å and A

§5. Investigation of Special Classes of Boundary Value Problems of the Type A in the Case O ≤ n ≤ m-1

§6. On the Conditions of Correctness of Problem A

§7. Solution of Problem A by Means of Two-Dimensional Integral Equations. Application of the Generalized Principle of Symmetry. Generalized Schwarz Integral

§8. The Boundary Value Problem of Inclined Derivative for an Elliptic Equation of the Second Order

§9. Application of Two-Dimensional Singular Integral Equations to the Boundary Value Problems

§10. Remarks Concerning Certain Papers on Problem A. Some Formulations of More General Problems

Appendix to Chapter IV. (B. Bojarski). On Special Cases of the Riemann-Hilbert Problem

Part Two Some Applications to Problems of the Theory of Surfaces and the Membrane Theory of Shells

Chapter V. Foundations of the General Theory of Infinitesimal Bendings of Surfaces

§1. Equations of Infinitesimal Bendings in Vectorial Form

§2. Equation of Infinitesimal Bendings with Respect to a Cartesian Coordinate System. The First Proof of Rigidity of Ovaloids

§3. The System of Equations for the Components of the Displacement Field in an Arbitrary Coordinate System on the Surface. Some Criteria of Rigidity

§4. On a Property of Surfaces of the Second Order

§5. The Rotation Field. The Characteristic Equation of Infinitesimal Bendings

§6. Bending Fields. Static Field

§7. Variations of Various Geometric Quantities Under Infinitesimal Bendings of the Surface. Some Criteria of Rigidity

§8. Conjunction Conditions on the Contact Lines. Some Criteria of Rigidity of Surface with Edges. Bush Constraints. Perfect Clamping

§9. Some Classes of Rigid Closed Sectionally Regular Surfaces

§10. Some Classes of Rigid Convex Surfaces with Edges

§11. Infinitesimal Bendings of Surfaces of Revolution

Chapter VI. Problems of the Membrane Theory of Shells

§1. Forces and Moments Due to the Stress Field

§2. Basic System of Equilibrium Equations of a Shell

§3. System of Equations of the Membrane State of Stress of Shells. Geometric Interpretation

§4. New Derivation of the Characteristic Equation

§5. Conditions of Existence of State (T). Boundary Value Problems

References

## Product details

- No. of pages: 698
- Language: English
- Copyright: © Pergamon 1962
- Published: January 1, 1962
- Imprint: Pergamon
- eBook ISBN: 9781483184678

## About the Author

### I. N. Vekua

## About the Editors

### I. N. Sneddon

### S. Ulam

### M. Stark

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