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AuthorMรผller, Claus. author
TitleAnalysis of Spherical Symmetries in Euclidean Spaces [electronic resource] / by Claus Mรผller
ImprintNew York, NY : Springer New York : Imprint: Springer, 1998
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Descript VIII, 226 p. online resource


This book gives a new and direct approach into the theories of special functions with emphasis on spherical symmetry in Euclidean spaces of arยญ 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 quanยญ 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


1 Notations and Basic Theorems> -- 1 The General Theory 9 -- ยง2 Primitive Spaces -- ยง3 The Completeness -- ยง4 The Funk-Hecke Formula -- ยง5 Representations and Interpolations -- ยง6 Homogeneous Harmonics -- 2 The Specific Theories -- ยง7 The Legendre Polynomials -- ยง8 The Laplace Integrals -- ยง9 The Gegenbauer Polynomials -- ยง10 The Associated Legendre Functions -- ยง1 The Associated Spaces yjn(q) -- ยง12 Harmonic Differential Operators -- ยง13 Maxwellโ{128}{153}s Theory of Multipoles -- 3 Spherical Harmonics and Differential Equations -- ยง14 The Laplace-Beltrami Operators -- ยง15 Spherical Harmonics as Eigenfunctions -- ยง16 The Legendre Differential Equation -- ยง17 The Legendre Functions as Hypergeometric Functions -- 4 Analysis on the Complex Unit Spheres -- ยง18 Homogeneous Harmonics in ?q -- ยง19 Invariant Integrals on S*q-1 -- ยง20 Complexification of the Funk-Hecke Formula -- ยง21 An Alternative System of Legendre Functions -- 5 The Bessel Functions -- ยง22 Regular Bessel Functions -- ยง23 Regular Hankel Functions -- ยง24 Recursive and Asymptotic Relations -- ยง25 Addition Formulas for Hankel Functions of Order Zero -- ยง26 Exponential Integrals with Bessel Functions -- ยง27 The Traditional Notations -- 6 Integral Transforms -- ยง28 Fourier Integrals -- ยง29 The Fourier Representation Theorem -- ยง30 The Parseval Identity -- ยง31 Examples -- 7 The Radon Transform -- ยง32 Radon Transforms and Fourier Transforms -- ยง33 Radon Transforms and Spherical Symmetries -- ยง34 The Nicholson Formulas -- 8 Appendix -- ยง35 The ?-Function. -- ยง36 The Hypergeometric Function -- ยง37 Elementary Asymptotics -- References

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