In preparing this translation for publication certain minor modifications and additions have been introduced into the original Russian text, in order to increase its readibility and usefulness. Thus, instead of the first person, the third person has been used throughout; wherever possible footnotes have been included with the main text. The chapters and their subsections of the Russian edition have been renamed parts and chapters respectively and the last have been numbered consecutively. An authors and subject index has been added. In particular, the former has been combined with the list of references of the original text, in order to enable the reader to find quickly all information on anyone reference in which he may be especially interested. This has been considered most important with a view to the difficulties experienced outside Russia in obtaining references, published in that country. Russian names have been printed in Russian letters in the authors index, in order to overcome any possible confusion arising from transliteration.

Table of Contents:
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.