Symmetry and separation of variables / Willard Miller, Jr. ; with a foreword by Richard Askey.

Format:

Book

Author/Creator:

Miller, Willard, author.

Contributor:

Askey, Richard, writer of foreword.

Series:

Encyclopedia of mathematics and its applications ; v. 4.

Encyclopedia of mathematics and its applications ; volume 4

Language:

English

Subjects (All):

Symmetry (Physics).

Functions, Special.

Differential equations, Partial--Numerical solutions.

Differential equations, Partial.

Separation of variables.

Physical Description:

1 online resource (xxx, 285 pages) : digital, PDF file(s).

Other Title:

Symmetry & Separation of Variables

Place of Publication:

Cambridge : Cambridge University Press, 1984.

Language Note:

English

Summary:

Originally published in 1977, this volume is concerned with the relationship between symmetries of a linear second-order 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. Some group-theoretic twists in the ancient method of separation of variables that can be used to provide a foundation for much of special function theory are shown. In particular, it is shown explicitly that all special functions that arise via separation of variables in the equations of mathematical physics can be studied using group theory.

Contents:

Cover; Half Title; Series Page; Title; Copyright; Contents; Editor's Statement; Foreword; References; Preface; CHAPTER 1 The Helmholtz Equation; 1.0 Introduction; 1.1 The Symmetry Group of the Helmholtz Equation; 1.2 Separation of Variables for the Helmholtz Equation; 1.3 Expansion Formulas Relating Separable Solutions; 1.4 Separation of Variables for the Klein-Gordon Equation; 1.5 Expansion Formulas for Solutions of the Klein-Gordon Equation; 1.6 The Complex Helmholtz Equation; 1.7 Weisner's Method for the Complex Helmholtz Equation; Exercises; CHAPTER 2 The Schrödinger and Heat Equations

3. Parabolic Cylindrical System4. Elliptic Cylindrical System; 6. Prolate Spheroidal System; 7. Oblate Spheroidal System; 8. Parabolic System; 9. Paraboloidal System; 10. Ellipsoidal System; 3.3 Lamè Polynomials and Functions on the Sphere; 3.4 Expansion Formulas for Separable Solutions of the Helmholtz Equation; 3.5 Non-Hilbert Space Models for Solutions of the Helmholtz Equation; 3.6 The Laplace Equation Δ3Ψ=0; 3.7 Identities Relating Separable Solutions of the Laplace Equation; Exercises; CHAPTER 4 The Wave Equation; 4.1 The Equation Ψ11-Δ2Ψ=0; 4.2 The Laplace Operator on the Sphere

7. The Lauricella Functions8. Mathieu Functions; APPENDIX C Elliptic Functions; REFERENCES; Subject Index

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Bibliography: p. 275-280.

ISBN:

1-139-88606-1

1-107-10228-6

1-107-08746-5

1-107-09975-7

1-107-09369-4

1-107-32562-5