Modern Mathematical Methods for Physicists and Engineers

This 763 page book, available as both hardback and paperback, is unique in the sense that it sails across the ocean of contemporary advanced mathematics by an impressive tour de force. In twelve ambitious chapters it covers major topics in set theory, groups, rings and fields, vector spaces, matrix theory, linear functionals and special functions. The depth of coverage, however, is highly uneven. Two chapters on linear functionals, one chapter on inner products and norms, and one chapter on convergence in normed vector spaces contain a wealth of finely tuned knowledge. On the other hand, Chapter 1 on the foundations of computation, and the final Chapter 12 on special functions, just skim the surface. The selection of material and its presentation is evidently aimed at computer scientists and computer engineers. Their colleagues in other disciplines will find little that applies directly to their respective fields in spite of the many (more than 400) exercise problems. The author's approach is fundamentally different from similar titles in the literature, e.g. the time-honoured multi-edition text for physicists by Arfken. It provides a cogent, elegant and often esoteric structural framework for mathematical formulations, but with limited interest in exploring detailed solutions. A case in point is the brief presentation of Weber's Y (or N) function, better known as the Bessel function of the second kind. It appears as an eigenfunction of the Bessel operator (Section 12.4), but without any mention of how this function can be routinely eliminated from solutions where the physical quantity of interest is finite at the origin - a cornerstone of numerous practical problem solutions in physics and engineering. Other omissions are equally vexing. In Chapter 3 (Evaluation of Functions), the bisection method is dismissed in nine lines, and no mention is made of the secant method and Wegstein's method, which are just as important for numerical root finding as the analytical (and temperamental) Newton's method. Classical integration techniques are discussed, but not even one of the modern methods (e.g. Gaussian, Chebyshev, Rodau, Lobatto, Laguerre, Hermite and Filon integration) is given any mention. The classical Euler method for integrating ordinary differential equations is the subject of four pages, but there is nothing at all on the widely used Runge-Kutta techniques, nor the highly efficient Richardson's extrapolation method. Modified Bessel functions and Kelvin functions, essential tools in physics and engineering, are completely missing. Fourier transforms are bestowed a short section (11.8), but Laplace transforms? - nothing. Complex numbers pop up here and there, but complex calculus is ignored. The list continues.... Will this work become `... an ideal textbook for senior undergraduate and graduate students in the physical sciences and engineering...' and `... a valuable reference for working engineers...'? The author's claim for the affirmative is extremely tenuous, inasmuch as only highly motivated students with a broad command of mathematics would be able (if willing) to master its contents. The working engineer would most likely look for pragmatic texts, which provide practical/numerical examples in equally modern packing (e.g. Advanced Engineering Mathematics by Robert J Lopez (Addison Wesley, 2000)). Finally, a number of witty comments throughout the book lighten the task of its reading. Some readers will consider the statement (p 494) that `... eigenvalue is a mongrel word...' (in contrast, presumably, with the purebred Eigenwert and valeur propre) to be amusingly clever. An entire Section 8.2 (albeit consisting only of four lines) tells us that `... there seem to be almost as many ways to spell Chebyshev as there are written human languages...', then gracefully provides the proper Cyrillic spelling. Thomas Z Fahidy