Boundary Value Problems
Boundary Value Problems : International Series of Monographs in Pure and Applied Mathematics
1st Edition
Published on 10. July 2014
584 pages
978-1-4831-6498-4 (ISBN)
Description
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 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.
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David L. Powers
Boundary Value Problems
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Content
Foreword to the First EditionForeword to the Second EditionIntroductionI. 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 III. 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.
