By Claus Müller

ISBN-10: 1461268273

ISBN-13: 9781461268277

This booklet provides a brand new and direct process into the theories of distinct services with emphasis on round symmetry in Euclidean areas of ar bitrary dimensions. crucial elements can even be referred to as straight forward end result of the selected innovations. The relevant subject is the presentation of round harmonics in a concept of invariants of the orthogonal crew. H. Weyl was once one of many first to indicate 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 used to be supported via many physicists. those rules are the major topic all through this treatise. whilst R. Richberg and that i begun this undertaking we have been shocked, how effortless and chic the final idea should be. one of many highlights of this publication is the extension of the classical result of round harmonics into the advanced. this is often relatively vital for the complexification of the Funk-Hecke formulation, that's effectively used to introduce orthogonally invariant recommendations of the lowered wave equation. The radial components of those ideas are both Bessel or Hankel capabilities, which play a massive position within the mathematical idea of acoustical and optical waves. those theories usually require a close research of the asymptotic habit of the options. The provided creation of Bessel and Hankel features yields at once the top phrases of the asymptotics. Approximations of upper order might be deduced.

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2) that the series in Lemma 2 satisfies for r E [0,1] q-2~ -2- L. 11) L rn cosncp = 00 n=l 1 +r 1_ r2 2 2 - rcoscp which is Lemma 2 with t = coscp and Pn(2jcoscp) = cosncp The Poisson identity is interesting because it provides a second proof of the completeness of the spherical harmonics. 12) r , (1 + r2 . 13) 2. 14) (1 + r2 - 2rt)~ < [(1 - r)2 + 2r(1 - t)]~ 1- r2 [2r(1 - to)]~ for t E [-1, to]. We now prove Lemma 3: Suppose f E C(Sq-1). 16) IS}-ll ~ E Sq-1 hq-l Gr(q; ~ . 17)dS('7) 1 = 1 but for [;q- 1, which does not contain the set {171~ .

17(q-1) ) 58 2. The Specific Theories because the sum over j yields N~~~~;I) Pm{q - 1; ~(q-l) addition theorem in (q - 1) dimensions. We set u : = ~(q-l) . 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.

Lemma 3: For q 2: 3 and 0 < t ~ 1 we have As a corollary to Lemma 1 we find Lemma 4: The Pn(q; t) satisfy for all t and q 2: 2 (1 - t2)p~(q; t) = -(n + q - 2) (Pn+1 (q; t) - tPn(q; t)) Exercise 1: Prove Lemma 4. Hint: Differentiate under the integral sign. §9 The Gegenbauer Polynomials The Legendre polynomials Pn(t) := Pn(3; t) were originally introduced as coefficients of the expansion 1 -:-(1-+-r-=-2---2r-t-:-C)1:-;jC::-2 00 = ~ rn Pn (t) which is valid in r E [0, 1) and t E [-1, 1]. The function on the left is the gravitational potential 1 Ix -yl of a unit mass at y.