By Claus Müller
This publication offers 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 components can even be referred to as undemanding as a result of selected innovations. The critical subject is the presentation of round harmonics in a conception of invariants of the orthogonal workforce. 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 career with quan tum mechanics and used to be supported by way of many physicists. those rules are the prime subject all through this treatise. whilst R. Richberg and that i began this undertaking we have been stunned, how effortless and chic the final conception may be. one of many highlights of this publication is the extension of the classical result of round harmonics into the advanced. this can be relatively very important for the complexification of the Funk-Hecke formulation, that is effectively used to introduce orthogonally invariant options of the lowered wave equation. The radial elements of those options are both Bessel or Hankel capabilities, which play an enormous function within the mathematical idea of acoustical and optical waves. those theories frequently require a close research of the asymptotic habit of the recommendations. The offered advent of Bessel and Hankel capabilities yields at once the major phrases of the asymptotics. Approximations of upper order might be deduced.
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Extra resources for Analysis of Spherical Symmetries in Euclidean Spaces
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. 42 2. The Specific Theories Theorem 1 will be referred to as the Laplace representation of the Legendre polynomials. To demonstrate its usefulness we derive another recurrence formula for the Pn(q; t) as well as some estimates. 8) Next, we prove inequalities for Pn(q; t).
We close this chapter with a formulation of Maxwell's theory of multipoles. §7 The Legendre Polynomials The classical theory of Legendre polynomials rests on the differential operator C. Müller, Analysis of Spherical Symmetries in Euclidean Spaces © Springer-Verlag New York, Inc. 1998 38 2. The Specific Theories This operator is self-adjoint because we have for /,9 E C(2)[-1,1]. The polynomial D(q)tPk(qjt) is of degree k, and we have for 1 = 0, 1, ... ) to all polynomials of degree less than k.
Theorem 1: Each space Yn(q), q 2: 3, n 2: 0 is the orthogonal direct sum of the associated spaces Y;;'(q) Proof: First, we show that the associated spaces in Yn(q) are mutually orthogonal. For 0 :0:::; k,m :0:::; n, Yk(q - l;'),Ym (q -1;·) and the scalar products < , >(q) ; < , >(q-1) we have (§11. ) = < Yk, Ym >(q-1) 1 >(q) A~(q; t) A;;:(q;t)(l- t2)~dt Thus Y! 1 Y;;' for k -=I- m. Moreover A~ is a bijection of Yk (q - 1) onto y~(q). 9) u=O N(q,n)u n l+u (1 - U)q-1 l+u 1 (1 - U)q-2 1 - u §11 The Associated Spaces y~ (q) 57 Since we are dealing with two linear spaces of equal dimension N(q, n), the assertion is proved.
Analysis of Spherical Symmetries in Euclidean Spaces by Claus Müller