Arscott F.M. Periodic Differential Equations: An Introduction to Mathieu, Lame, and Allied Functions

Pergamon Press, 1964. — 283 p. — ISBN: 008009984X, 9780080099842.Of recent years, the two subjects of special functions and ordinary linear differential equations have (in this country at least) lain somewhat in the penumbra of a partial eclipse. Many mathematicians, not without reason, have come to regard the former as little more than a haphazard collection of ugly and unmemorable formulae, while attention in the latter has been directed mostly to existence-theorems and similar results for equations of general type. Only rarely does one find mention,at post-graduate level, of any problems in connection with the process of actually solving such equations. The electronic computer may perhaps be partly to blame for this, since the impression prevails in many quarters that almost any differential equation problem can be merely "put on the machine", so that finding an analytic solutions largely a waste of time. This, however, is only a small part of the truth, for at the higher levels there are generally so many parameters or boundary conditions involved that numerical solutions, even if practicable, give no real idea of the properties of the equation. Moreover, any analyst of sensibility will feel that to fall back on numerical techniques savours somewhat of breaking a door with a hammer when one could, with a little trouble, find the key.Periodic Differential Equations: An Introduction to Mathieu, Lamé, and Allied Functions covers the fundamental problems and techniques of solution of periodic differential equations. This book is composed of 10 chapters that present important equations and the special functions they generate, ranging from Mathieu's equation to the intractable ellipsoidal wave equation.This book starts with a survey of the main problems related to the formation of periodic differential equations. The subsequent chapters deal with the general theory of Mathieu's equation, Mathieu functions of integral order, and the principles of asymptotic expansions. These topics are followed by discussions of the stable and unstable solutions of Mathieu's general equation; general properties and characteristic exponent of Hill's equation; and the general nature and solutions of the spheroidal wave equation. The concluding chapters explore the polynomials, orthogonality properties, and integral relations of Lamé's equation. These chapters also describe the wave functions and solutions of the ellipsoidal wave equation.This book will prove useful to pure and applied mathematicians and functional analysis.