Talk:Associated Legendre function/Addendum

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Revision as of 10:13, 12 July 2009 by imported>Paul Wormer
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I added a proof of orthogonality and a derivation of the normalization constant for the first equation in the Orthogonality relations section in on the main page. Dan Nessett 16:42, 11 July 2009 (UTC)

Comments on proof

1. The proof starts out by implicitly proving the anti-Hermiticity of

Indeed, let w(x) be a function with w(1) = w(−1) = 0, then

Hence

The latter result is used in the proof given in the Addendum.

2. When as an intermediate the ordinary Legendre polynomials Pl are introduced, we may use a result from the theory of orthogonal polynomials. Namely, a Legendre polynomial of order l is orthogonal to any polynomial of lower order. We meet (kl)

then

The bra is a polynomial of order k, and since kl, the bracket is non-zero only if k = l.

Then, knowing this, the hard work (given in the Addendum) of computing the normalization constant remains.

--Paul Wormer 15:13, 12 July 2009 (UTC)