Miller W. Symmetry and Separation of Variables

New York: Addison-Wesley Publishing Co., 1977. - 291p.This book is concerned with the relationship between the symmetries of a
linear partial differential equation of mathematical physics, the
coordinate systems in which the equation admits solutions via separation
of variables, and the properties of the special functions that arise in
this manner. It is shown how the method of separation of variables can be
used to provide a group-theoretic foundation for much of special function
theory. Special functions included in this treatment are the functions of
Mathieu, Ince, Lame and others. A group-theoretic machine can be
constructed which describes the various separable coordinate systems and
expansion theorems relating distinct separable solutions. Indeed, for the
most important linear equations the separated solutions can be
characterized as eigenfunctions of a set of commuting second-order
symmetric operators in the enveloping algebra. The problem of expanding
one set of separable solutions in terms of another reduces to a problem in
the representation theory of the Lie symmetry algebra. This book
constitutes an important step in the group-theoretic approach to special
functions. It is clearly written and should be accessible to a broad
spectrum of readers.The Helmholtz Equation
The Schrödinger and Heat Equations
The Three-Variable Helmholtz and Laplace Equations
The Wave Equation
The Hypergeometric Function and Its Generalizations
Lie Groups and Algebras
Basic Properties of Special Functions
Elliptic Functions