This book presents a new and original method for the solution of boundary value problems in angles for second-order elliptic equations with constant coefficients and arbitrary boundary operators. This method turns out to be applicable to many different areas of mathematical physics, in particular to diffraction problems in angles and to the study of trapped modes on a sloping beach.

Giving the reader the opportunity to master the techniques of the modern theory of diffraction, the book introduces methods of distributions, complex Fourier transforms, pseudo-differential operators, Riemann surfaces, automorphic functions, and the Riemann-Hilbert problem.

The book will be useful for students, postgraduates and specialists interested in the application of modern mathematics to wave propagation and diffraction problems.

Inhalt

Introduction. - Part I Survey of Diffraction Theory. - The Early Theory of Diffraction. - Fresnel-Kirchhoff Diffraction Theory. - Stationary and Time-Dependent Diffraction. - The Sommerfeld Theory of Diffraction by Half-Plane. - Diffraction byWedge After Sommerfeld's Article. - Part II Method of Automorphic Functions on Complex Characteristics. - Stationary Boundary Value Problems in Convex Angles. - Extension to the Plane. - Boundary Conditions via the Cauchy Data. - Connection Equation on the Riemann Surface. - On Equivalence of the Reduction. - Undetermined Algebraic Equations on the Riemann Surface. - Automorphic Functions on the Riemann Surface. - Functional Equation with a Shift. - Lifting to the Universal Covering. - The Riemann-Hilbert Problem on the Riemann Surface. - The Factorization. - The Saltus Problem and Final Formula. - The Reconstruction of Solution and the Fredholmness. - Extension of the Method to Non-convex Angle. - Comments.