< Talk:Associated Legendre functionRevision as of 10:13, 12 July 2009 by imported>Paul Wormer
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)
1. The proof starts out by implicitly proving the anti-Hermiticity of
![{\displaystyle \nabla _{x}\equiv {\frac {d}{dx}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f06d65f8c6a5771d11221c51773bc81758b5d574)
Indeed, let w(x) be a function with w(1) = w(−1) = 0, then
![{\displaystyle \langle wg|\nabla _{x}f\rangle =\int _{-1}^{1}w(x)g(x)\nabla _{x}f(x)dx=\left[w(x)g(x)f(x)\right]_{-1}^{1}-\int _{-1}^{1}{\Big (}\nabla _{x}w(x)g(x){\Big )}f(x)dx=-\langle \nabla _{x}(wg)|f\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab17d0257943f93ceb089f3ddd0ebb911fe723fd)
Hence
![{\displaystyle \nabla _{x}^{\dagger }=-\nabla _{x}\;\Longrightarrow \;\left(\nabla _{x}^{\dagger }\right)^{l+m}=(-1)^{l+m}\nabla _{x}^{l+m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4c402d0f80449994c9c7e8928bde5d938ee962)
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 (k ≤ l)
![{\displaystyle \langle w\nabla _{x}^{m}P_{k}|\nabla _{x}^{m}P_{l}\rangle \quad {\hbox{with}}\quad w\equiv (1-x^{2})^{m},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed47c0d91ddf1b1d9d1b0e74103df1fe7c78b3e1)
then
![{\displaystyle \langle w\nabla _{x}^{m}P_{k}|\nabla _{x}^{m}P_{l}\rangle =(-1)^{m}\langle \nabla _{x}^{m}(w\nabla _{x}^{m}P_{k})|P_{l}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf95337d724c6ab8b332a1d0686dc1501fcaabf3)
The bra is a polynomial of order k, and since k ≤ l, 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)