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. Language: English