# Boundary Value Problems

## 1st Edition

**Author:**F. D. Gakhov

**Editors:**I. N. Sneddon M. Stark S. Ulam

**eBook ISBN:**9781483164984

**Imprint:**Pergamon

**Published Date:**1st January 1966

**Page Count:**584

**View all volumes in this series:**International Series in Pure and Applied Mathematics

## Table of Contents

Foreword to the First Edition

Foreword to the Second Edition

Introduction

I. Integrals of the Cauchy Type

§ 1. Definition of the Cauchy Type Integral and Examples

§ 2. Functions Satisfying the Holder Condition

2.1. Definition and Properties

2.2. Functions of Many Variables

§ 3. Principal Value of the Cauchy Type Integral

3.1. Improper Integral

3.2. Principal Value of Singular Integral

3.3. Many-Valued Functions

3.4. Principal Value of Singular Curvilinear Integral

3.5. Properties of the Singular Integral

§ 4. Limiting Values of the Cauchy Type Integral over the Real Axis

4.1. The Basic Lemma

4.2. The Sokhotski Formula

4.3. The Conditions Ensuring That an Arbitrary Complex Function is the Boundary Value of a Function Analytic In the Domain

4.4. Limiting Values of the Derivatives. Derivatives of Limiting Values. Derivatives of a Singular Integral

4.5. The Sokhotski Formula for Corner Points of a Contour

4.6. Integrals of the Cauchy Type over the Real Axis

§ 5. Properties of the Limiting Values of the Cauchy Type Integral

5.1. The Limiting Values Satisfy the Holder Condition

5.2. Extension of the Assumptions

5.3. Some New Results

§ 6. The Hilbert Formula for the Limiting Values of the Real and Imaginary Parts of an Analytic Function

6.1. The Cauchy and Schwarz Kernels

6.2. The Hilbert Formula

§ 7. The Change of the Order of Integration in a Repeated Singular Integral

7.1. The Case When One Integral is Ordinary

7.2. The Transposition Formula

7.3. Inversion of the Singular Integral with the Cauchy Kernel for the Case of a Closed Contour

§ 8. Behavior of the Cauchy Type Integral at the Ends of the Contour of Integration and at the Points of Density Discontinuities

8.1. The Case of Density Satisfying the Holder Condition on L, Including the Ends

8.2. The Case of Discontinuity of the First Kind

8.3. The Particular Case of a Power Singularity

8.4. The General Case of a Power Singularity

8.5. Singularity of Logarithmic Type

8.6. Singularities of Power-Logarithmic Type

8.7. Integral of the Cauchy Type over a Complicated Contour

§ 9. Limiting Values of Generalized Integrals and Double Cauchy Integrals

9.1. Formulation of the Problem

9.2. Formula Analogous to the Sokhotski Formula for the Cauchy Type Integral

9.3. The Formula for the Change of the Order of Integration

9.4. Multiple Cauchy Integrals. Formulation of the Problem

9.5. Singular Double Integral. Poincaré-Bertrand Formula

9.6. Sokhotski's Formula

§ 10. Integral of the Cauchy Type and Potentials

§ 11. Historical Notes

Problems on Chapter I

II. Riemann Boundary Value Problem

§ 12. The Index

12.1. Definition and Basic Properties

12.2. Computation of the Index

§ 13. Some Auxiliary Theorems

§ 14. The Riemann Problem for a Simply-Connected Domain

14.1. Formulation of the Problem

14.2. Determination of Sectionally Analytic Function in Accordance with Given Jump

14.3. Solution of the Homogeneous Problem

14.4. The Canonical Function of the Homogeneous Problem

14.5. Solution of the Nonhomogeneous Problem

14.6. Examples

14.7. The Riemann Problem for the Semi-Plane

§ 15. Exceptional Cases of the Riemann Problem

15.1. The Homogeneous Problem

15.2. The Non-homogeneous Problem

§ 16. Riemann Problem for Multiply-Connected Domain. Some New Results

16.1. Formulation of the Problem

16.2. Solution of the Problem

16.3. Some New Results

§ 17. Riemann Boundary Value Problem with Shift

17.1. Formulation of the Problem and General Remarks

17.2. Problem with Zero Jump

17.3. Problem with Given Jump

17.4. The Homogeneous Problem with Zero Index

17.5. Reducing the Shift Problem to the Ordinary Riemann Problem

§ 18. Other Generalized Problems

18.1. Formulation of the Problems and Notation

18.2. Reduction to the Simplest Case

§ 19. Historical Notes

Problems on Chapter II

III. Singular Integral Equations with Cauchy Kernel

§ 20. Basic Concepts and Notation

20.1. Singular Integral Equation

20.2. Fundamental Results of the Theory of Fredholm Integral Equations

§ 21. The Dominant Equation

21.1. Reduction to the Riemann Boundary Value Problem

21.2. Solution of the Dominant Equation

21.3. The Solution of the Equation Adjoint to the Dominant Equation

21.4. Examples

21.5 . Approximate Solution

21.6. The Behavior of the Solution at Corner Points

§ 22. Regularization of the Complete Equation

22.1. Product of Singular Operators

22.2. Regularizing Operator

22.3. Methods of Regularization

22.4. Relation between Solutions of Singular and Regularized Equations

§ 23. Fundamental Properties of Singular Equations

23.1. Some Properties of Adjoint Operators

23.2. Fundamental Theorem on Singular Integral Equations (Noether's Theorems)

23.3. Some Corollaries

§ 24. Equivalent Regularization. The Third Method of Regularization

24.1. Statement of the Problem. Various Interpretations of the Concept of Equivalent Regularization

24.2. Equivalent Regularization of an Operator

24.3. Equivalent Regularization from the Left of the Singular Equation. The Necessary Condition

24.4. Conjugate Equation. Another Form of the Conditions of Solubility of the Non-Homogeneous Equation

24.5. Theorem on Equivalent Regularization of an Equation

24.6. Regularization by Solving the Dominant Equation (The Method of Carleman-Vekua)

24.7. Example

§ 25. Exceptional Cases of Singular Integral Equations

25.1. Solution of the Dominant Equation

25.2. Regularization of the Complete Equation

§ 26. Historical Notes

Problems on Chapter III

IV. Hilbert Boundary Value Problem and Singular Integral Equations with Hilbert Kernel

§ 27. Formulation of the Hilbert Problem and Some Auxiliary Formula

27.1. Formulation of the Hilbert Problem

27.2. The Schwarz Operator for Simply-Connected Domain

27.3. Determination of an Analytic Function Possessing a Pole, in Terms of the Value of Its Real Part on the Contour (Problem A)

§ 28. Regularizing Factor

28.1. Definition of the Regularizing Factor

28.2. Real Regularizing Factor

28.3. Regularizing Factor with Constant Modulus

28.4. Other Forms of the Regularizing Factor

§ 29. The Hilbert Boundary Value for Simply-Connected Domains

29.1. The Homogeneous Problem

29.2. The Non-Homogeneous Problem

29.3. Problem for the Unit Circle

29.4. The Hilbert Problem for Exterior Domain

29.5. Examples

§ 30. Relation between the Hilbert and Riemann Problems

30.1. Comparison of the Formula Representing the Solutions of the Boundary Value Problems

30.2. Connection between the Boundary Conditions

§ 31. Singular Integral Equation with Hilbert Kernel

31.1. Connection of the Dominant Equation with the Hilbert Boundary Value Problem

31.2. The Homogeneous Equation

31.3. The Non-homogeneous Equation

31.4. Equation with Constant Coefficients

31.5 The Complete Equation and Its Regularization

31.6. Basic Properties of Equation with Hilbert Kernel

§32. Boundary Value Problems for Polyharmonic and Polyanalytic Functions, Reducible to the Hilbert Boundary Value Problem

32.1. Representation of Polyharmonic and Polyanalytic Functions by Analytic Functions

32.2. Formulation of the Boundary Value Problems for Polyanalytic Functions

32.3. Boundary Value Problems for a Circle

32.4. Boundary Value Problems for Domains Mapped onto the Circle by Means of Rational Functions

32.5. Solution of the Basic Problem for the Theory of Elasticity for the Domain Bounded By Pascal's Limaçon

§33. The Inverse Boundary Value Problem for Analytic Functions

33.1. Formulation of the Problem

33.2. Solution of the Interior Problem

33.3. Other Methods of Prescribing Boundary Values

33.4. Solution of the Exterior Problem

33.5. The Number of Solutions of the Exterior Problem

33.6. The Schwarz Problem with a Logarithmic Singularity on the Contour

33.7. Singular Points of the Contour

33.8. The Single-Sheet Nature of the Solution

33.9. Some Further Topics

§33*. Historical Notes
Problems on Chapter IV
V. Various Generalized Boundary Value Problems
§ 34. Boundary Value Problem of Hilbert Type, with the Boundary Condition Containing Derivatives
34.1. Formulation of the Problem and Various Forms of the Boundary Conditions
34.2. Representation of an Analytic Function by a Cauchy Type Integral with a Real Density
34.3. The Cauchy Type Integral the Density of Which is the Product of a Given Complex Function and A Real Function
34.4. Integral Representation of An Analytic Function the mth Derivative of Which is Representable by the Cauchy Type Integral
34.5. Reduction of the Boundary Value Problem to a Fredholm Integral Equation
34.6, Problems of Solubility of the Boundary Value Problem
34.7. Other Methods of Investigation
§ 35. Boundary Value Problem of Riemann Type with the Boundary Condition Containing Derivatives
35.1. An Integral Representation for Sectionally Analytic Function
35.2. Solution of the Boundary Value Problem
35.3. New Method of Solving the Problem
35.4. Singular Integro-Differential Equation
§ 36. The Hilbert Boundary Value Problem for Multiply-Connected Domains
36.1. The Dirichlet Problem for Multiply-Connected Domains
36.2. The Schwarz Operator for a Multiply-Connected Domain
36.3. Problem A for a Multiply-Connected Domain
36.4. The Regularizing Factor
36.5. The Solution of the Homogeneous Hilbert Problem in the Class of Many-Valued Functions
36.6. Integral Equation of the Hilbert Problem
36.7. The Adjoint Integral Equation and the Adjoint Hilbert Problem
36.8. Investigation of the Solubility Problems
36.9. Investigation of the Cases X = 0 and X=M − 1
36.10. Connection with the Mapping onto a Plane with Slits
36.11. Concluding Remarks
36.12. Some New Results
§36*. Inverse Boundary Value Problem for a Multiply-Connected Domain

36

*.1. Formulation of the Problem*

36.2. Solution of the Interior Problem

36

36*.3. Solubility Conditions

§ 37. General Boundary Value Problem of Riemann Type for Multiply-Connected Domains

37.1. The Integral Representation

37.2. Boundary Value Problem and Integro-Differential Equation

§ 38. Boundary Value Problems for Equations of Elliptic Type

38.1. General Information

38.2. Classical Methods of Solution

38.3. Integral Representation of Solutions

38.4. The General Boundary Value Problem

§ 39. Boundary Value Problems for Systems of Elliptic Equations

39.1. Various Forms of the System and General Remarks

39.2. Functions of Class C

39.3. The Fundamental Solution

39.4. The Normal Form of the System

39.5. An Auxiliary Representation of the Solutions and Its Implications

39.6. Integral Representation of Solutions

39.7. Boundary Value Problems

39.8. Riemann Boundary Value Problem. Formulation and Auxiliary Relationships

39.9. The Solution of the Riemann Boundary Value Problem

39.10. Additional Remarks

§ 40. Historical Notes

Problems on Chapter V

VI. Boundary Value Problems and Singular Integral Equations with Discontinuous Coefficients and Open Contours

§ 41. Solution of the Riemann Problem with Discontinuous Coefficients by Reduction to a Problem with Continuous Coefficients

41.1. Formulation of the Problem and Determination of the Function in Terms of a Given Jump

41.2. Fundamental Auxiliary Functions

41.3. Reduction to a Problem with Continuous Coefficients. The Simplest Case

41.4. Solution of the Homogeneous Problem

41.5. Solution of the Non-Homogeneous Problem

§ 42. Riemann Boundary Value Problem for Open Contours

42.1. Formulation and Solution of the Problem

42.2. Example

42.3. Inversion of the Cauchy Type Integral

§ 43. Direct Solution of the Riemann Problem

43.1. The Problem for an Open Contour

43.2. The Problem with Discontinuous Coefficients

43.3. Examples

43.4. Concluding Remarks

§44. Riemann Problem for a Complicated Contour

44.1. Formulation of the Problem

44.2. A New Method of Solution of the Riemann Problem for a Closed Curve

44.3. The General Case

44.4. The Case of Coincidence of the Ends

§ 45. Exceptional Cases and the General Concept of Index

45.1. Introductory Remarks

45.2. The Homogeneous Riemann Problem

45.3. Exceptional Points at Contour Ends

§ 46. Hilbert Boundary Value Problem with Discontinuous Coefficients

46.1. Hilbert Problem for the Semi-Plane

46.2. Example

46.3. Mixed Boundary Value Problem for Analytic Functions

46.4. The Dirichlet Problem and Its Modifications for the Plane with Slits

§ 47. The Dominant Equation for Open Contours

47.1. Basic Concepts and Notation

47.2 Solution of the Dominant Equation

47.3. Solution of the Equation Adjoint to the Dominant Equation

47.4. Examples

§ 48. Complete Equation for Open Contours

48.1. Regularization by Solving the Dominant Equation

48.2. Investigation of the Regularized Equation

48.3. Other Methods of Regularization. Equivalent Regularization

48.4. Basic Properties of the Singular Equation

§ 49. The General Case

49.1. Equation on a Complex Contour and with Discontinuous Coefficients

49.2. Example

49.3. Exceptional Cases

49.4. Approximate Methods

§ 50. Historical Notes

Problem on Chapter VI

VII. Integral Equations Soluble in Closed Form

§ 51. Equations with Automorphic Kernels and a Finite Group

51.1. Some Results of the Theory of Finite Groups of Linear Fractional Transformations and of Automorphic Functions

51.2. Reduction of a Complete Singular Equation to a Boundary Value Problem

51.3. Solution of the Boundary Value Problem

51.4. Solution of the Integral Equation

51.5. Example

51.6. The Case When the Auxiliary Analytic Function Does Not Vanish at Infinity

51.7. Additional Remarks

§ 52. Continuation. The Case of an Infinite Group

52.1. The Riemann Boundary Value Problem

52.2. The Singular Integral Equation

52.3. Some Applications

52.4. Integral Equation with a Non-Fundamental Automorphic Function

52.5. Example

§ 53. Some Types of Integral Equations with Power and Logarithmic Kernels

53.1. Abel Integral Equation

53.2. Integral with a Power Kernel

53.3. The Generalized Abel Integral Equation

53.4. Integral with a Logarithmic Kernel

53.5. Integral Equations with Logarithmic Kernels

53.6. Various Possible Generalizations

§ 54. Historical Notes

Problems on Chapter VII

References

Index

Other Titles in the Series

## Description

Boundary Value Problems is a translation from the Russian of lectures given at Kazan and Rostov Universities, dealing with the theory of boundary value problems for analytic functions.

The emphasis of the book is on the solution of singular integral equations with Cauchy and Hilbert kernels. Although the book treats the theory of boundary value problems, emphasis is on linear problems with one unknown function. The definition of the Cauchy type integral, examples, limiting values, behavior, and its principal value are explained. The Riemann boundary value problem is emphasized in considering the theory of boundary value problems of analytic functions. The book then analyzes the application of the Riemann boundary value problem as applied to singular integral equations with Cauchy kernel. A second fundamental boundary value problem of analytic functions is the Hilbert problem with a Hilbert kernel; the application of the Hilbert problem is also evaluated. The use of Sokhotski's formulas for certain integral analysis is explained and equations with logarithmic kernels and kernels with a weak power singularity are solved. The chapters in the book all end with some historical briefs, to give a background of the problem(s) discussed.

The book will be very valuable to mathematicians, students, and professors in advanced mathematics and geometrical functions.

## Details

- No. of pages:
- 584

- Language:
- English

- Copyright:
- © Pergamon 1966

- Published:
- 1st January 1966

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483164984