Generalized Analytic Functions

¿ForewordPart 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.