By Claus Müller

This publication supplies a brand new and direct procedure into the theories of targeted capabilities with emphasis on round symmetry in Euclidean areas of ar bitrary dimensions. crucial elements may also be referred to as easy a result of selected options. The vital subject is the presentation of round harmonics in a idea of invariants of the orthogonal staff. H. Weyl was once one of many first to show that round harmonics needs to be greater than a lucky wager to simplify numerical computations in mathematical physics. His opinion arose from his career with quan tum mechanics and was once supported via many physicists. those principles are the best subject all through this treatise. whilst R. Richberg and that i began this venture we have been shocked, how effortless and chic the final concept might be. one of many highlights of this e-book is the extension of the classical result of round harmonics into the complicated. this is often really vital for the complexification of the Funk-Hecke formulation, that is effectively used to introduce orthogonally invariant suggestions of the diminished wave equation. The radial components of those suggestions are both Bessel or Hankel capabilities, which play an immense position within the mathematical idea of acoustical and optical waves. those theories frequently require a close research of the asymptotic habit of the strategies. The offered advent of Bessel and Hankel capabilities yields at once the prime phrases of the asymptotics. Approximations of upper order may be deduced.

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Note: We need only the leading coefficient a~(q) explicitly in the following calculations. ) 2 The Specific Theories The preceding chapter has described the general properties of the spaces Yn(q), but there is a need to fill the general frame with concrete and explicit data. We have seen that the Legendre polynomials contain all that is necessary to determine an orthonormal system of spherical harmonics. This chapter is therefore devoted to the Pn(q;·) and their many relations and possibilities.

13) to Lemma 2: For t, s, u E [-1,1] and q to by the 3 we have ~ ~) N(q,n)p (. Isq-11 ~ . T'/(q-l») nq,st+uVl-s~vl-t- = ISql_21 N(q - 1, m)A~(q, t)A~(q, s)Pn(q - 1; u) We multiply by Pdq -1; u)(l- U2)~ on both sides and integrate over [-1,1]. 17) We have now gained a rather closed and explicit theory of complete systems of functions on spheres, and it is an interesting question if this knowledge can be used in other situations. So far, we used almost exclusively the standard polar coordinates to formulate our results.

An element of the space Y~(q). 3) for x = XqCq + X(q-l) ,X(q-l) E IRq-I. sQ-2 (x q + iX(q_l) . A*TJ(q_l»)ndsq - 2 (TJ) ISq1_21 } 8Q- r (X q 2 + iX(q_l) . «(q_l»)nds q - 2 ( ) Ln(x) because the substitution «(q-l) = A*TJ(q-l) does not change the value of the integral. 3) we have also Ln(cq) = 1. Thus Ln is the Legendre harmonic of degree n and dimension q. From Lemma 2, §7 we get where ~(q-l) is any point of Sq-2 and t E [-1,1]. Evaluation ofthe integral in standard coordinates now yields Laplace's first integral.