Re: Associated Legendre Definition

*To*: mathgroup at smc.vnet.net*Subject*: [mg25723] Re: Associated Legendre Definition*From*: leko at ix.netcom.com (J. Leko)*Date*: Thu, 19 Oct 2000 04:35:48 -0400 (EDT)*References*: <8se9qf$6q3@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <8se9qf$6q3 at smc.vnet.net>, Preben Bohn <pmib at my-deja.com> wrote: > In Mathematica, the associated Legendre polynomial is defined as > > P(n,m,x) = (-1)^m (1-x^2)^(m/2) d^m/dx^m (P(n,x)) > > while in Schaum's Outlines 'Mathematical Handbook of Formulas and > tables' it is defined as > > P(n,m,x) = (1-x^2)^(m/2) d^m/dx^m (P(n,x)) > >What is true (or doesn't it matter)? In a matter of speaking, both are correct. You are seeing two types of notation: Mathematica is following the one favored by atomic spectra and electrodynamics textbooks (see chapter 3 of J. D. Jackson's Classical Electrodynamics, 2nd edition, 1975). Your Schaum's Outline follows the one used in Arfken's Mathematical Methods for Physicists (3rd edition, 1985). The factor (-1)^m is a phase factor, usually referred to as the Condon-Shortley phase. Its effect is to introduce an alternation of sign among the positive m spherical harmonics (Arfken; p. 682). It seems that the publishers of the Schaum's Outline share Arfken's idea that "This (-1)^m seems an unnecessary complication... It will be included in the definition of the spherical harmonics." (see footnote on page 668). As long as it is present in either the Plm or the Ylm expression (but not in both), the expressions will be OK. Mayra Martinez