Singular Integral Equations

I Fundamental Propkrtibs of Cauchy Integrals.- 1 The Holder Condition.- 1 Smooth and piecewise smooth lines.- 2 Some properties of smooth lines.- 3 The Holder Condition (H condition).- 4 Generalization to the case of several variables.- 5 Two auxiliary inequalities.- 6 Sufficient conditions for the H condition to be satisfied.- 7 Sufficient conditions for the H condition to be satisfied (continued).- 8 Sufficient conditions for the H condition to be satisfied (continued).- 2 Integrals of the Cauehy type.- 9 Definitions.- 10 The Cauehy integral.- 11 Connection with logarithmic potential. Historical remarks.- 12 The values of Cauehy integrals on the path of integration.- 13 The tangential derivative of the potential of a simple layer.- 14 Sectionally continuous functions.- 15 Sectionally holomorphic functions.- 16 The limiting value of a Cauehy integral.- 17 The Plemelj formulae.- 18 Generalization of the formulae for the difference in limiting values.- 19 The continuity behaviour of the limiting values.- 20 The continuity behaviour of the limiting values (continued).- 21 On the behaviour of the derivative of a Cauehy integral near the boundary.- 22 On the behaviour of a Cauehy integral near the boundary.- 3 Some corollaries on Cauehy integrals.- 23 Poincare-Bertrand tranformation formula.- 24 On analytic continuation of a function given on the boundary of a region.- 25 Generalization of Harnack's theorem.- 26 On sectionally holomorphic functions with discontinuities (case of contours).- 27 Inversion of the Cauehy integral (case of contours).- 28 The Hilbert inversion formulae.- 4 Cauehy integrals near ends of the line of integration.- 29 Statement of the principal results.- 30 An auxiliary estimate.- 31 Deduction of formula (29.5).- 32 Deduction of formula (29.8).- 33 On the behaviour of a Cauehy integral near points of discontinuity.- II The Hilbert and the Biemann-Helbert Problems and Singular Integral Equations (Case of Contours).- 5 The Hilbert and Riemann-Hilbert boundary problems.- 34 The homogeneous Hilbert problem.- 35 General solution of the homogeneous Hilbert problem. The Index.- 36 Associate homogeneous Hilbert problems.- 37 The non-homogeneous Hilbert problem.- 38 On the extension to the whole plane of analytic functions given on a circle or half-plane.- 39 The Riemann-Hilbert problem.- 40 Solution of the Riemann-Hilbert problem for the circle.- 41 Example. The Dirichlet problem for a circle.- 42 Reduction of the general case to that of a circular region.- 43 The Riemann-Hilbert problem for the half-plane.- 6 Singular integral equations with Cauehy type kernels (case of contours).- 44 Singular equations and singular operators.- 45 Fundamental properties of singular operators.- 46 Adjoint operators and adjoint equations.- 47 Solution of the dominant equation.- 48 Solution of the equation adjoint to the dominant equation.- 49 Some general remarks.- 50 On the reduction of a singular integral equation.- 51 On the reduction of a singular integral equation (continued).- 52 On the resolvent of the Fredholm equation.- 53 Fundamental theorems.- 54 Real equations.- 55 I. N. Vekua's theorem of equivalence. An alternative proof of the fundamental theorems.- 56 Comparison of a singular integral equation with a Fredholm equation. The Quasi-Fredholm singular equation. Reduction to the canonical form.- 57 Method of reduction, due to T. Carleman and I. N. Vekua.- 58 Introduction of the parameter ?.- 59 Brief remarks on some other results.- III Applications to Some Boundary Problems.- 7 The Dirichlet problem.- 60 Statement of the Dirichlet and the modified Dirichlet problem. Uniqueness theorems.- 61 Solution of the modified Dirichlet problem by means of the potential of a double layer.- 62 Some corollaries.- 63 Solution of the Dirichlet problem.- 64 Solution of the modified Dirichlet problem, using the modified potential of a simple layer.- 65 Solution of the Dirichlet problem by the potential of a simple layer. Fundamental problem of electrostatics.- 8 Various representations of holomorpkic functions by Cauehy and analogous integrals.- 66 General remarks.- 67 Representation by a Cauehy integral with real or imaginary density.- 68 Representation by a Cauehy integral with density of the form (a + ib) ?.- 69 Integral representation by I. N. Vekua.- 9 Solution of the generalized Riemann-Hilbert-Poincare problem.- 70 Preliminary remarks.- 71 The generalized Riemann-Hilbert-Poincare problem (Problem V). Reduction to an integral equation.- 72 Investigation of the solubility of Problem V.- 73 Criteria of solubility of Problem V.- 74 The Poineare problem (Problem P).- 75 Examples.- 76 Some generalizations and applications Singular Integral Equations 1.- IV The Hilbert Problem in the Case of Arcs or Discontinuous Boundary Conditions and Some of its Applications.- 10 The Hilbert problem in the case of arcs or discontinuous boundary conditions.- 77 Definitions.- 78 Definition of a sectionally holomorphic function for a given discontinuity.- 79 The homogeneous Hilbert problem for open contours.- 80 The associate homogeneous Hilbert problem. Associate classes.- 81 Solution of the non-homogeneous Hilbert problem for arcs.- 82 The concept of the class h of functions given on L.- 83 Some generalizations.- 84 Examination of the problem ?+ + ?? = g.- 85 The Hilbert problem in the case of discontinuous coefficients.- 86 The Hilbert problem in the case of discontinuous coefficients (continued).- 87 Connection with the case of arcs.- 11 Inversion formulae for arcs.- 88 The inversion of a Cauchy integral.- 89 Some variations of the inversion problem.- 90 Some variations of the inversion problem (continued).- 12 Effective solution of some boundary problems of the theory of harmonic functions.- 91 The Dirichlet and analogous problems for the plane with cuts distributed along a straight line.- 92 The Dirichlet and analogous problems for the plane with cuts distributed over a circle.- 98 The Riemann-Hilbert problem for discontinuous coefficients.- 94 Particular cases: The mixed problem of the theory of holomorphic functions.- 95 The mixed problem for the half-plane. Formula of M. V. Keldysh and L. I. Sedov.- 13 Effective solution of the principal problems of the static theory of elasticity for the half-plane, circle and analogous regions.- 96 General formulae of the plane theory of elasticity.- 97 The first, second and mixed boundary problems for an elastic half-plane.- 49 Some general remarks.- 50 On the reduction of a singular integral equation.- 51 On the reduction of a singular integral equation (continued).- 52 On the resolvent of the Fredholm equation.- 53 Fundamental theorems.- 54 Real equations.- 55 I. N. Vekua's theorem of equivalence. An alternative proof of the fundamental theorems.- 56 Comparison of a singular integral equation with a Fredholm equation. The Quasi-Fredholm singular equation. Reduction to the canonical form.- 57 Method of reduction, due to T. Carleman and I. N. Vekua.- 58 Introduction of the parameter ?.- 59 Brief remarks on some other results.- III Applications to Some Boundary Problems.- 7 The Dirichlet problem.- 60 Statement of the Dirichlet and the modified Dirichlet problem. Uniqueness theorems.- 61 Solution of the modified Dirichlet problem by means of the potential of a double layer.- 62 Some corollaries.- 63 Solution of the Dirichlet problem.- 64 Solution of the modified Dirichlet problem, using the modified potential of a simple layer.- 65 Solution of the Dirichlet problem by the potential of a simple layer. Fundamental problem of electrostatics.- 8 Various representations of holomorpkic functions by Cauehy and analogous integrals.- 66 General remarks.- 67 Representation by a Cauehy integral with real or imaginary density.- 68 Representation by a Cauehy integral with density of the form (a + ib) ?.- 69 Integral representation by I. N. Vekua.- 9 Solution of the generalized Riemann-Hilbert-Poincare problem.- 70 Preliminary remarks.- 71 The generalized Riemann-Hilbert-Poincare problem (Problem V). Reduction to an integral equation.- 72 Investigation of the solubility of Problem V.- 73 Criteria of solubility of Problem V.- 74 The Poineare problem (Problem P).- 75 Examples.- 76 Some generalizations and applications Singular Integral Equations 1.- IV The Hilbert Problem in the Case of Arcs or Discontinuous Boundary Conditions and Some of its Applications.- 10 The Hilbert problem in the case of arcs or discontinuous boundary conditions.- 77 Definitions.- 78 Definition of a sectionally holomorphic function for a given discontinuity.- 79 The homogeneous Hilbert problem for open contours.- 80 The associate homogeneous Hilbert problem. Associate classes.- 81 Solution of the non-homogeneous Hilbert problem for arcs.- 82 The concept of the class h of functions given on L.- 83 Some generalizations.- 84 Examination of the problem ?+ + ?? = g.- 85 The Hilbert problem in the case of discontinuous coefficients.- 86 The Hilbert problem in the case of discontinuous coefficients (continued).- 87 Connection with the case of arcs.- 11 Inversion formulae for arcs.- 88 The inversion of a Cauchy integral.- 89 Some variations of the inversion problem.- 90 Some variations of the inversion problem (continued).- 12 Effective solution of some boundary problems of the theory of harmonic functions.- 91 The Dirichlet and analogous problems for the plane with cuts distributed along a straight line.- 92 The Dirichlet and analogous problems for the plane with cuts distributed over a circle.- 98 The Riemann-Hilbert problem for discontinuous coefficients.- 94 Particular cases: The mixed problem of the theory of holomorphic functions.- 95 The mixed problem for the half-plane. Formula of M. V. Keldysh and L. I. Sedov.- 13 Effective solution of the principal problems of the static theory of elasticity for the half-plane, circle and analogous regions.- 96 General formulae of the plane theory of elasticity.- 97 The first, second and mixed boundary problems for an elastic half-plane.- 98 The problem of pressure of rigid stamps on the boundary of an elastic half-plane in the absence of friction.- 99 The problem of pressure of rigid stamps on the boundary of an elastic half-plane in the absence of friction (continued).- 100 Equilibrium of a rigid stamp on the boundary of an elastic half- plane in the presence of friction.- 101 Another method of solution of the boundary problem for the half- plane.- 102 The problem of contact of two elastic bodies (the generalized plane problem of Hertz).- 103 The fundamental boundary problems for the plane with straight cuts.- 104 The boundary problems for circular regions.- 105 Certain analogous problems. Generalizations.- V Singular Integral Equations for the Case of Arcs or Discontinuous Coefficients and Some of their Applications.- 14 Singular integral equations for the case of arcs and continuous coefficients.- 106 Definitions.- 107 Solution of the dominant equation.- 108 Solution of the equation adjoint to the dominant equation.- 109 Reduction of the singular equation K? = f.- 110 Reduction of the singular equation K? = g.- 111 Investigation of the equation resulting from the reduction.- 112 Solution of a singular equation. Fundamental theorems.- 113 Application to the dominant equation of the first kind.- 114 Reduction and solution of an equation of the first kind.- 115 An alternative method for the investigation of singular equations.- 15 Singular integral equations in the case of discontinuous coefficients.- 116 Definitions.- 117 Reduction and solution of singular equations in the case of discontinuous coefficients.- 16 Application to the Dirichlet problem and similar problems.- 118 The Dirichlet and similar problems for the plane, cut along arcs of arbitrary shape.- 119 Reduction to a Fredhohn equation. Examples.- 120 The Dirichlet problem for the plane, cut along a finite number of arcs of arbitrary shape.- 17 Solution of the intgro-differential-equation of the theory of aircraft wings of finite span.- 121 The mtegro-differential equation of the theory of aircraft wings of finite span.- 122 Reduction to a regular Fredholm equation.- 25 Generalization of Harnack's theorem.- 26 On sectionally holomorphic functions with discontinuities (case of contours).- 27 Inversion of the Cauehy integral (case of contours).- 28 The Hilbert inversion formulae.- 4 Cauehy integrals near ends of the line of integration.- 29 Statement of the principal results.- 30 An auxiliary estimate.- 31 Deduction of formula (29.5).- 32 Deduction of formula (29.8).- 33 On the behaviour of a Cauehy integral near points of discontinuity.- II The Hilbert and the Biemann-Helbert Problems and Singular Integral Equations (Case of Contours).- 5 The Hilbert and Riemann-Hilbert boundary problems.- 34 The homogeneous Hilbert problem.- 35 General solution of the homogeneous Hilbert problem. The Index.- 36 Associate homogeneous Hilbert problems.- 37 The non-homogeneous Hilbert problem.- 38 On the extension to the whole plane of analytic functions given on a circle or half-plane.- 39 The Riemann-Hilbert problem.- 40 Solution of the Riemann-Hilbert problem for the circle.- 41 Example. The Dirichlet problem for a circle.- 42 Reduction of the general case to that of a circular region.- 43 The Riemann-Hilbert problem for the half-plane.- 6 Singular integral equations with Cauehy type kernels (case of contours).- 44 Singular equations and singular operators.- 45 Fundamental properties of singular operators.- 46 Adjoint operators and adjoint equations.- 47 Solution of the dominant equation.- 48 Solution of the equation adjoint to the dominant equation.- 49 Some general remarks.- 50 On the reduction of a singular integral equation.- 51 On the reduction of a singular integral equation (continued).- 52 On the resolvent of the Fredholm equation.- 53 Fundamental theorems.- 54 Real equations.- 55 I. N. Vekua's theorem of equivalence. An alternative proof of the fundamental theorems.- 56 Comparison of a singular integral equation with a Fredholm equation. The Quasi-Fredholm singular equation. Reduction to the canonical form.- 57 Method of reduction, due to T. Carleman and I. N. Vekua.- 58 Introduction of the parameter ?.- 59 Brief remarks on some other results.- III Applications to Some Boundary Problems.- 7 The Dirichlet problem.- 60 Statement of the Dirichlet and the modified Dirichlet problem. Uniqueness theorems.- 61 Solution of the modified Dirichlet problem by means of the potential of a double layer.- 62 Some corollaries.- 63 Solution of the Dirichlet problem.- 64 Solution of the modified Dirichlet problem, using the modified potential of a simple layer.- 65 Solution of the Dirichlet problem by the potential of a simple layer. Fundamental problem of electrostatics.- 8 Various representations of holomorpkic functions by Cauehy and analogous integrals.- 66 General remarks.- 67 Representation by a Cauehy integral with real or imaginary density.- 68 Representation by a Cauehy integral with density of the form (a + ib) ?.- 69 Integral representation by I. N. Vekua.- 9 Solution of the generalized Riemann-Hilbert-Poincare problem.- 70 Preliminary remarks.- 71 The generalized Riemann-Hilbert-Poincare problem (Problem V). Reduction to an integral equation.- 72 Investigation of the solubility of Problem V.- 73 Criteria of solubility of Problem V.- 74 The Poineare problem (Problem P).- 75 Examples.- 76 Some generalizations and applications Singular Integral Equations 1.- IV The Hilbert Problem in the Case of Arcs or Discontinuous Boundary Conditions and Some of its Applications.- 10 The Hilbert problem in the case of arcs or discontinuous boundary conditions.- 77 Definitions.- 78 Definition of a sectionally holomorphic function for a given discontinuity.- 79 The homogeneous Hilbert problem for open contours.- 80 The associate homogeneous Hilbert problem. Associate classes.- 81 Solution of the non-homogeneous Hilbert problem for arcs.- 82 The concept of the class h of functions given on L.- 83 Some generalizations.- 84 Examination of the problem ?+ + ?? = g.- 85 The Hilbert problem in the case of discontinuous coefficients.- 86 The Hilbert problem in the case of discontinuous coefficients (continued).- 87 Connection with the case of arcs.- 11 Inversion formulae for arcs.- 88 The inversion of a Cauchy integral.- 89 Some variations of the inversion problem.- 90 Some variations of the inversion problem (continued).- 12 Effective solution of some boundary problems of the theory of harmonic functions.- 91 The Dirichlet and analogous problems for the plane with cuts distributed along a straight line.- 92 The Dirichlet and analogous problems for the plane with cuts distributed over a circle.- 98 The Riemann-Hilbert problem for discontinuous coefficients.- 94 Particular cases: The mixed problem of the theory of holomorphic functions.- 95 The mixed problem for the half-plane. Formula of M. V. Keldysh and L. I. Sedov.- 13 Effective solution of the principal problems of the static theory of elasticity for the half-plane, circle and analogous regions.- 96 General formulae of the plane theory of elasticity.- 97 The first, second and mixed boundary problems for an elastic half-plane.- 98 The problem of pressure of rigid stamps on the boundary of an elastic half-plane in the absence of friction.- 99 The problem of pressure of rigid stamps on the boundary of an elastic half-plane in the absence of friction (continued).- 100 Equilibrium of a rigid stamp on the boundary of an elastic half- plane in the presence of friction.- 101 Another method of solution of the boundary problem for the half- plane.- 102 The problem of contact of two elastic bodies (the generalized plane problem of Hertz).- 103 The fundamental boundary problems for the plane with straight cuts.- 104 The boundary problems for circular regions.- 105 Certain analogous problems. Generalizations.- V Singular Integral Equations for the Case of Arcs or Discontinuous Coefficients and Some of their Applications.- 14 Singular integral equations for the case of arcs and continuous coefficients.- 106 Definitions.- 107 Solution of the dominant equation.- 108 Solution of the equation adjoint to the dominant equation.- 109 Reduction of the singular equation K? = f.- 110 Reduction of the singular equation K? = g.- 111 Investigation of the equation resulting from the reduction.- 112 Solution of a singular equation. Fundamental theorems.- 113 Application to the dominant equation of the first kind.- 114 Reduction and solution of an equation of the first kind.- 115 An alternative method for the investigation of singular equations.- 15 Singular integral equations in the case of discontinuous coefficients.- 116 Definitions.- 117 Reduction and solution of singular equations in the case of discontinuous coefficients.- 16 Application to the Dirichlet problem and similar problems.- 118 The Dirichlet and similar problems for the plane, cut along arcs of arbitrary shape.- 119 Reduction to a Fredhohn equation. Examples.- 120 The Dirichlet problem for the plane, cut along a finite number of arcs of arbitrary shape.- 17 Solution of the intgro-differential-equation of the theory of aircraft wings of finite span.- 121 The mtegro-differential equation of the theory of aircraft wings of finite span.- 122 Reduction to a regular Fredholm equation.- 123 Certain generalizations.- VI The Hilbert Problem for Several Unknown Functions and Systems of Singular Integral Equations.- 18 The Hilbert problem for several unknown functions.- 124 Definitions.- 125 Auxiliary theorems.- 126 The homogeneous Hilbert problem.- 127 The fundamental system of solutions of the homogeneous Hilbert problem and its general solution.- 128 The non-homogeneous Hilbert problem.- 129 Supplement to the solution of a dominant system of singular integral equations and of its associate system.- 19 Systems of singular integral equations with Cauchy type kernels and some supplements.- 130 Definitions. Auxiliary theorems.- 131 Reduction of a system of singular equations. Fundamental theorems.- 132 Other methods of reduction and the investigation of systems of singular equations.- 133 Brief remarks regarding important generalizations and supplements.- Appendix 1 On smooth and piecewise smooth lines.- Appendix 2 On the behaviour of the Cauchy integral near corner points.- Appendix 3 An elementary proposition regarding bi-orthogpnal systems of functions.- References and author index.