Singular Integral Equations

This work focuses on the distributional solutions of singular integral equations, progressing from basic concepts of the classical theory to the more difficult two-dimensional problems.

1 Reference Material.- 1.1 Introduction.- 1.2 Singular Integral Equations.- 1.3 Improper Integrals.- 1.3.1 The Gamma function.- 1.3.2 The Beta function.- 1.3.3 Another important improper integral.- 1.3.4 A few integral identities.- 1.4 The Lebesgue Integral.- 1.5 Cauchy Principal Value for Integrals.- 1.6 The Hadamard Finite Part.- 1.7 Spaces of Functions and Distributions.- 1.8 Integral Transform Methods.- 1.8.1 Fourier transform.- 1.8.2 Laplace transform.- 1.9 Bibliographical Notes.- 2 Abel''s and Related Integral Equations.- 2.1 Introduction.- 2.2 Abel''s Equation.- 2.3 Related Integral Equations.- 2.4 The equation $$\int_{0}^{s} {{{{(s - t)}}^{\beta }}g(t)dt = f(s), \Re e \beta > - 1}$$.- 2.5 Path of Integration in the Complex Plane.- 2.6 The Equation $$\int_{{{{C}_{{a\xi }}}}} {\frac{{g(z)dz}}{{{{{(z - \xi )}}^{\nu }}}}} + k\int_{{{{C}_{{\xi b}}}}} {\frac{{g(z)dz}}{{{{{(\xi - z)}}^{\nu }}}}} = f(\xi )$$.- 2.7 Equations on a Closed Curve.- 2.8 Examples.- 2.9 Bibliographical Notes.- 2.10 Problems.- 3 Cauchy Type Integral Equations.- 3.1 Introduction.- 3.2 Cauchy Type Equation of the First Kind.- 3.3 An Alternative Approach.- 3.4 Cauchy Type Equations of the Second Kind.- 3.5 Cauchy Type Equations on a Closed Contour.- 3.6 Analytic Representation of Functions.- 3.7 Sectionally Analytic Functions (z?a)n?v(z?b)m+v.- 3.8 Cauchy''s Integral Equation on an Open Contour.- 3.9 Disjoint Contours.- 3.10 Contours That Extend to Infinity.- 3.11 The Hilbert Kernel.- 3.12 The Hilbert Equation.- 3.13 Bibliographical Notes.- 3.14 Problems.- 4 Carleman Type Integral Equations.- 4.1 Introduction.- 4.2 Carleman Type Equation over a Real Interval.- 4.3 The Riemann-Hilbert Problem.- 4.4 Carleman Type Equations on a Closed Contour.- 4.5 Non-Normal Problems.- 4.6 A Factorization Procedure.- 4.7 An Operational Approach.- 4.8 Solution of a Related Integral Equation.- 4.9 Bibliographical Notes.- 4.10 Problems.- 5 Distributional Solutions of Singular Integral Equations.- 5.1 Introduction.- 5.2 Spaces of Generalized Functions.- 5.3 Generalized Solution of the Abel Equation.- 5.4 Integral Equations Related to Abel''s Equation.- 5.5 The Fractional Integration Operators ??.- 5.6 The Cauchy Integral Equation over a Finite Interval.- 5.7 Analytic Representation of Distributions of ?''[a, b].- 5.8 Boundary Problems in A[a,b].- 5.9 Disjoint Intervals.- 5.9.1 The problem [RjF]j =hj.- 5.9.2 The equation A1?1(0F) + A2?2(F) = G.- 5.10 Equations Involving Periodic Distributions.- 5.11 Bibliographical Notes.- 5.12 Problems.- 6 Distributional Equations on the Whole Line.- 6.1 Introduction.- 6.2 Preliminaries.- 6.3 The Hilbert Transform of Distributions.- 6.4 Analytic Representation.- 6.5 Asymptotic Estimates.- 6.6 Distributional Solutions of Integral Equations.- 6.7 Non-Normal Equations.- 6.8 Bibliographical Notes.- 6.9 Problems.- 7 Integral Equations with Logarithmic Kernels.- 7.1 Introduction.- 7.2 Expansion of the Kernel In |x-y|.- 7.3 The Equation $$\int_{a}^{b} {\ln } \left| {x - y} \right|g(y)dy = f(x)$$.- 7.4 Two Related Operators.- 7.5 Generalized Solutions of Equations with Logarithmic Kernels.- 7.6 The Operator $$\int_{a}^{b} {(P(x - y)\ln \left| {x - y} \right| + Q(x,y))g(y)dy}$$.- 7.7 Disjoint Intervals of Integration.- 7.8 An Equation Over a Semi-Infinite Interval.- 7.9 The Equation of the Second Kind Over a Semi-Infinite Interval.- 7.10 Asymptotic Behavior of Eigenvalues.- 7.11 Bibliographical Notes.- 7.12 Problems.- 8 Wiener-Hopf Integral Equations.- 8.1 Introduction.- 8.2 The Holomorphic Fourier Transform.- 8.3 The Mathematical Technique.- 8.4 The Distributional Wiener-Hopf Operators.- 8.5 Illustrations.- 8.6 Bibliographical Notes.- 8.7 Problems.- 9 Dual and Triple Integral Equations.- 9.1 Introduction.- 9.2 The Hankel Transform.- 9.3 Dual Equations with Trigonometric Kernels.- 9.4 Beltrami''s Dual Integral Equations.- 9.5 Some Triple Integral Equations.- 9.6 Erdélyi-Köber Operators.- 9.7 Dual Integral Equations of the Titchmarsh Type.- 9.8 D