Excerpt from An Abstract Formulation of the Method of Separation of Variables The method of separation of variables is one of the few methods by which partial differential equations can be solved explicitly. The method, however, is obviously limited to those cases where the partial differential equation expressed in a suitable coordinate system, has solutions which are products of functions of the separate coordinates; consequently, much research has been devoted to the determination of the coordinate systems in which parti cular equations, such as the wave equation, have separable Solutions. Along with this, there has been a study of the special functions defined by the sep arate differential equations. However, even when the separable coordinat systems and the special functions needed have been determined, there is still one question to be discussed, namely the question of the type of boundary con ditions for which the method of separation of variables can be used. To answer this question, it was found convenient to place it and the whole method of sep aration of variables in an abstract framework. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.