Near/far-side angular decompositions of Legendre polynomials using the amplitude-phase method

Near/far-side angular decompositions of Legendre polynomials using the amplitude-phase method A decomposition of Legendre polynomials into propagating angular waves is derived with the aid of an amplitude-phase method. This decomposition is compared with the ’Nussenzveig/Fuller’ so called near/far-side decomposition of Legendre polynomials. The latter decomposition requires the Legendre function of the second kind. This is not the case with the amplitude-phase decomposition. Both representations have the same asymptotic expressions for large values of $$(l+1/2)\sin \theta $$ ( l + 1 / 2 ) sin θ , where l and $$\theta $$ θ are the polynomial degree and the angle respectively. Furthermore, both components of both representations satisfy the Legendre differential equation. However, we show the two representations are not identical. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Chemistry Springer Journals

Near/far-side angular decompositions of Legendre polynomials using the amplitude-phase method

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by The Author(s)
Subject
Chemistry; Physical Chemistry; Theoretical and Computational Chemistry; Math. Applications in Chemistry
ISSN
0259-9791
eISSN
1572-8897
D.O.I.
10.1007/s10910-017-0752-x
Publisher site
See Article on Publisher Site

Abstract

A decomposition of Legendre polynomials into propagating angular waves is derived with the aid of an amplitude-phase method. This decomposition is compared with the ’Nussenzveig/Fuller’ so called near/far-side decomposition of Legendre polynomials. The latter decomposition requires the Legendre function of the second kind. This is not the case with the amplitude-phase decomposition. Both representations have the same asymptotic expressions for large values of $$(l+1/2)\sin \theta $$ ( l + 1 / 2 ) sin θ , where l and $$\theta $$ θ are the polynomial degree and the angle respectively. Furthermore, both components of both representations satisfy the Legendre differential equation. However, we show the two representations are not identical.

Journal

Journal of Mathematical ChemistrySpringer Journals

Published: May 5, 2017

References

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