By Claus Müller

This booklet offers a brand new and direct strategy into the theories of exact features with emphasis on round symmetry in Euclidean areas of ar bitrary dimensions. crucial components will also be referred to as straightforward end result of the selected suggestions. The primary subject is the presentation of round harmonics in a idea of invariants of the orthogonal team. H. Weyl used to be one of many first to show that round harmonics has to be greater than a lucky bet to simplify numerical computations in mathematical physics. His opinion arose from his profession with quan tum mechanics and was once supported by way of many physicists. those principles are the prime subject matter all through this treatise. while R. Richberg and that i began this venture we have been shocked, how effortless and stylish the final concept can be. one of many highlights of this booklet is the extension of the classical result of round harmonics into the advanced. this is often fairly very important for the complexification of the Funk-Hecke formulation, that's effectively used to introduce orthogonally invariant strategies of the lowered wave equation. The radial components of those recommendations are both Bessel or Hankel capabilities, which play an enormous function within the mathematical concept of acoustical and optical waves. those theories frequently require an in depth research of the asymptotic habit of the ideas. The awarded advent of Bessel and Hankel features yields at once the major phrases of the asymptotics. Approximations of upper order should be deduced.

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We limit the parameters of the functions to integral degrees, orders, and dimensions, which is sufficient for a description of the phenomena of orthogonal invariance. As was already mentioned, the associated functions are needed for explicit computations with spherical harmonics. r(q+2j -1) - r(~+j) (2n+q-2)r(n+q+j-2) 54 2. (2n + q - 2)(n + q + j - 3)! We now introduce the normed associated functions. ] 1/2 pj( . r(9) n q, (n+q-3)! r(~) 2q - 2(n + q + j - 3)! ). 18) = 3 we get simple expressions (n + ~)(n - j)!

One such branch is Gegenbauer's theory of the coefficients 00 1 _ ""' 2 ) -~r ( 1 + r - 2rt v n=O v (). ncnt,VEll" 1Ill+ §9 The Gegenbauer Polynomials 45 which we now regard as a generalization of the Laplace integrals [12]. We start with Definition 1: For v E 1R+, n E No the polynomial CV(t) : = n (n + 2v - 1) r(v + ~) 1+1 (t + isJ1-="t2)n(1 _ s2t-1ds y'1rr(V)-l n with the binomial coefficient (n+~-l) is called the Gegenbauer polynomial of degree n and index v. 2) (n + 2vn - 1) z n = (1 _ 1Z)2V; Izl < C~(t).

T) 2q-2 n ! n q, §1l The Associated Spaces y~(q) 55 §11 The Associated Spaces y~ (q) There are two aspects of the subject of this section. We may see it as a process to generate orthonormalized bases in a rational way, but we may also regard it in view of symmetry properties, related to the isotropical group J(q, eq). We follow the historical way and generalize Laplace's first integral. 1) a(q) = eq . 2) homogeneous harmonics, depending on 71(q-l). 3) is a homogenous and harmonic polynomial of degree n, and consequently an element of y~(q).