Canonical problems in scattering and potential theory. Part 1: Canonical structures in potential theory (Q2734574)

This book is the first part of a work on scattering, the object of which, according to the preface is to discuss analytic and semi-analytic techniques for ``determining the diffraction from structures comprising edges and other complex cavity features.'' Generally speaking the problems discussed involve a partial differential equation with mixed boundary conditions, involving the solution of dual equations and where the concept of regularization is used in solving the problem. It is a substantial work and consequently the best thing is to give a chapter by chapter survey, pointing out particular features of interest.NEWLINENEWLINENEWLINEThe first chapter is entitled Laplace's equation and discusses its form in several coordinate systems. In addition there is a classification of various solution methods for dual equations, and of Abel transforms. The second chapter, entitled series and integral equations introduces topics such as Jacobi polynomials, the relation between series and integral equations, and triple series equations. In the third chapter the authors consider open spherical shells with problems such as spherical caps and barrels. The importance of Bouwkamp's inversion theorem is pointed out. Unusually, and to be welcomed consequently, is a treatment of some problems involving superconductive shells.NEWLINENEWLINENEWLINEThe next three chapters deal with electrostatic problems associated with respectively spheroidal shells, toroidal shells and conial structures. Examples of the structures considered are spheroidal conductors with holes and slots, toroidal shells with various shaped slots, non-coplanar infinite strips, finite cones and frustra, bicones and hollow spindles.NEWLINENEWLINENEWLINEThe seventh chapter is concerned with two-dimensional potential theory. Amongst the problems treated are circular arcs, systems of thin strips and axially slotted elliptic cylinders: There is also a short account of how slotted cylinders of arbitrary shape may be dealt with. The eighth chapter considers more complicated structures. These include flat plates, elliptic and polygonal, and finite plates and a discussion of a capacitor composed of a spherical cap and a circular disc.NEWLINENEWLINENEWLINEFollowing on the main text are a number of appendices which deal with topics such as notation, special functions, elements of functional analysis, transforms and integration of series. There is an up to date list of 79 references, nearly all in English, but some in German and Russian, the latest in fact still being in process of publication.NEWLINENEWLINENEWLINEThe text is supplemented by a number of calculations associated with the methods suggested. In many cases, the results of 11-term approximations to the infinite term equations used are given, and the authors suggest that this is satisfactory. The printing is good and no errors were noted. The price, considering the length of the book, and the large amount of mathematical analysis involved is reasonable. The authors can be congratulated on producing such a scholarly work and the book can thoroughly be recommended.