Analysis of Spherical Symmetries in Euclidean Spaces

1 Notations and Basic Theorems>.- 1 The General Theory 9.- §2 Primitive Spaces.- §3 The Completeness.- §4 The Funk-Hecke Formula.- §5 Representations and Interpolations.- §6 Homogeneous Harmonics.- 2 The Specific Theories.- §7 The Legendre Polynomials.- §8 The Laplace Integrals.- §9 The Gegenbauer Polynomials.- §10 The Associated Legendre Functions.- §1 The Associated Spaces yjn(q).- §12 Harmonic Differential Operators.- §13 Maxwell’s Theory of Multipoles.- 3 Spherical Harmonics and Differential Equations.- §14 The Laplace-Beltrami Operators.- §15 Spherical Harmonics as Eigenfunctions.- §16 The Legendre Differential Equation.- §17 The Legendre Functions as Hypergeometric Functions.- 4 Analysis on the Complex Unit Spheres.- §18 Homogeneous Harmonics in ?q.- §19 Invariant Integrals on S*q-1.- §20 Complexification of the Funk-Hecke Formula.- §21 An Alternative System of Legendre Functions.- 5 The Bessel Functions.- §22 Regular Bessel Functions.- §23 Regular Hankel Functions.- §24 Recursive and Asymptotic Relations.- §25 Addition Formulas for Hankel Functions of Order Zero.- §26 Exponential Integrals with Bessel Functions.- §27 The Traditional Notations.- 6 Integral Transforms.- §28 Fourier Integrals.- §29 The Fourier Representation Theorem.- §30 The Parseval Identity.- §31 Examples.- 7 The Radon Transform.- §32 Radon Transforms and Fourier Transforms.- §33 Radon Transforms and Spherical Symmetries.- §34 The Nicholson Formulas.- 8 Appendix.- §35 The ?-Function..- §36 The Hypergeometric Function.- §37 Elementary Asymptotics.- References.