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This self-contained book offers a new and direct approach to the theories of special functions with emphasis on spherical symmetry in Euclidean spaces of arbitrary dimensions. Based on many years of lecturing to mathematicians, physicists and engineers in scientific research institutions in Europe and the USA, the author uses elementary concepts to present the spherical harmonics in a theory of invariants of the orthogonal group. One of the highlights is the extension of the classical results of the spherical harmonics into the complex - particularly important for the complexification of the Funk-Hecke formula which successfully leads to new integrals for Bessel- and Hankel functions with many applications of Fourier integrals and Radon transforms. Numerous exercises stimulate mathematical ingenuity and bridge the gap between well-known elementary results and their appearance in the new formations.

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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.

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