Overview
- Self-contained work *
- Much material published here for the first time * Uses elementary concepts of the theory of invariants of orthogonal groups and harmonics
- Results treated in an appendix to avoid disrupting the course of the ideas
Part of the book series: Applied Mathematical Sciences (AMS, volume 129)
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About this book
This book gives a new and direct approach into the theories of special functions with emphasis on spherical symmetry in Euclidean spaces of ar bitrary dimensions. Essential parts may even be called elementary because of the chosen techniques. The central topic is the presentation of spherical harmonics in a theory of invariants of the orthogonal group. H. Weyl was one of the first to point out that spherical harmonics must be more than a fortunate guess to simplify numerical computations in mathematical physics. His opinion arose from his occupation with quan tum mechanics and was supported by many physicists. These ideas are the leading theme throughout this treatise. When R. Richberg and I started this project we were surprised, how easy and elegant the general theory could be. One of the highlights of this book is the extension of the classical results of spherical harmonics into the complex. This is particularly important for the complexification of the Funk-Hecke formula, which is successfully used to introduce orthogonally invariant solutions of the reduced wave equation. The radial parts of these solutions are either Bessel or Hankel functions, which play an important role in the mathematical theory of acoustical and optical waves. These theories often require a detailed analysis of the asymptotic behavior of the solutions. The presented introduction of Bessel and Hankel functions yields directly the leading terms of the asymptotics. Approximations of higher order can be deduced.
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Table of contents (9 chapters)
Authors and Affiliations
Bibliographic Information
Book Title: Analysis of Spherical Symmetries in Euclidean Spaces
Authors: Claus Müller
Series Title: Applied Mathematical Sciences
DOI: https://doi.org/10.1007/978-1-4612-0581-4
Publisher: Springer New York, NY
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eBook Packages: Springer Book Archive
Copyright Information: Springer Science+Business Media New York 1998
Hardcover ISBN: 978-0-387-94949-9Published: 20 November 1997
Softcover ISBN: 978-1-4612-6827-7Published: 11 October 2012
eBook ISBN: 978-1-4612-0581-4Published: 06 December 2012
Series ISSN: 0066-5452
Series E-ISSN: 2196-968X
Edition Number: 1
Number of Pages: VIII, 226
Topics: Analysis, Geometry, Math. Applications in Chemistry, Theoretical, Mathematical and Computational Physics