Boundary Value Problems
- 1st Edition - January 1, 1966
- Imprint: Pergamon
- Author: F. D. Gakhov
- Editors: I. N. Sneddon, M. Stark, S. Ulam
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 3 2 5 6 - 3
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 6 4 9 8 - 4
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… Read more
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Request a sales quoteBoundary 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.
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*.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
- Edition: 1
- Published: January 1, 1966
- No. of pages (eBook): 584
- Imprint: Pergamon
- Language: English
- Paperback ISBN: 9781483132563
- eBook ISBN: 9781483164984
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