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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract
This paragraph gives a survey of the aspects, theorems, and techniques of the analysis in ℝq in a notation which is adapted to the special problems of spherical symmetry. The classical concepts of the tensor calculus and the formalisms of the theory of differential forms are both used as we go along, the results are stated, but no proofs of general theorems are presented because several good books devoted to the subject are available.
Claus Müller

1. The General Theory

Abstract
The concept of invariance with respect to transformations of a group is one of the most important and successful ideas of nineteenth century mathematics. After the use of coordinates had dominated many branches of mathematics and physics for centuries, a critical review of these methods was initiated by a new look on its foundations.
Claus Müller

2. The Specific Theories

Abstract
The preceding chapter has described the general properties of the spaces Y n (q), but there is a need to fill the general frame with concrete and explicit data. We have seen that the Legendre polynomials contain all that is necessary to determine an orthonormal system of spherical harmonics. This chapter is therefore devoted to the Pn(q; ·) and their many relations and possibilities. This part contains only a selection of identities and special results. An explicit orthogonal basis of yn (q) was found by Laplace for q = 3. His discovery can be easily extended to higher dimensions. We add a description of the isotropically invariant associated spaces.
Claus Müller

3. Spherical Harmonics and Differential Equations

Abstract
When Laplace and Legendre began their investigations of mathematical physics, this was in a major way a research in differential equations. The Laplacean Δ(3) was dominant and the necessity to find many solutions of Δ(3)U = 0 led to the method of separation of variables, which reduced the three-dimensional problems to three intertwined one-dimensional problems.
Claus Müller

4. Analysis on the Complex Unit Spheres

Abstract
The last chapter showed the variety of methods and results that can be obtained when the range of definitions is extended to the complex domain.
Claus Müller

5. The Bessel Functions

Abstract
We now turn to the solutions of the differential equations
$$({{\Delta }_{{(q)}}} + \lambda )U = 0$$
Claus Müller

6. Integral Transforms

Abstract
Linear mappings of function spaces in ℝq by means of integral transforms have become increasingly important for many applications of the mathematical sciences. The great success of the theories, commonly termed as Fourier analysis, is only one example of this development. Global integral transforms in ℝq with relevance in mathematical physics, probability theory, or geometry have nearly always a property, which may be called “preservation of spherical symmetry.”
Claus Müller

7. The Radon Transform

Abstract
The Radon transform, which was first discussed in 1912 by J. Radon, can be seen as a special case of a symmetry-preserving integral transform. The theory of this transformation is closely connected to Fourier transforms. The name Radon transform was first used by F. John in 1955. Its use in computer tomography aroused a renaissance of interest in some of the older problems in optics and geometrical probabilities. But the discoveries were not confined to Euclidean spaces. The present state is well documented [15], [19].
Claus Müller

8. Appendix

Abstract
For x ∈ ℝ+ the Г-function is defined as
$$ \Gamma (x): = \int_0^\infty {t^{x - 1} e^{ - t} dt} $$
(§35.1)
and we find for the derivatives (k ∈ ℕ)
$$ {{\left( {\frac{d}{{dx}}} \right)}^{k}}\Gamma (x) = \smallint _{0}^{\infty }{{(\ln t)}^{k}}{{t}^{{x - 1}}}{{e}^{{ - t}}}dt$$
(§35.2)
because differentiation and integration may be interchanged.
Claus Müller

Backmatter

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