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...