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Re: Associated Legendre Definition

In article <8se9qf$6q3 at>, Preben Bohn <pmib at> 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

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