212 pages | 2007 | PDF | 13 Mb
Book Description:
Ordinary linear equations, with which this book deals, have always attracted particular attention by their comparative tractability and their numerous practical applications. Extensive monographs have been devoted to many separate branches of the theory, such as spherical and cylindrical harmonics, expansions in series of ortho gonal functions, oscillation and comparison theorems, the Heaviside calculus, polyhedral, elliptic modular and automorphic functions. While some branches arose out of physical problems, others were created by the progress of the theory of functions and of the theory of groups. Many important ideas were first worked out in connexion with the hypergeometric equation by Euler, Gauss, Kummer, Rie mann, or Schwarz, and were then generalized by Fuchs, Klein, Poincare, and many other writers of the highest distinction. The present Introduction is based on lectures to senior under graduates at Oxford, and is designed for students who have already taken an elementary course of differential equations, but have not yet specialized in one of the more advanced branches. It is not a compendium of this vast subject to which no single author could do justice, but a selection of investigations of moderate length and difficulty, illustrating those aspects of it which are most familiar to myself. The first five chapters deal with properties common to wide classes of equations, and the last five are devoted to a more detailed examination of the hypergeometric equation, Laplaces linear equation, and the equations of Lame and Mathieu. I have not discussed systematically the equations of Legendre and Bessel, as there are so many admirable accounts of them in English suitable for students of every grade