Singular Integral Equations: Boundary Problems of Functions Theory and Their Applications to Mathematical Physics (Paperback, 1958)

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 Dirichle