Associated Legendre Polynomials - Reparameterization in Terms of Angles

Reparameterization in Terms of Angles

These functions are most useful when the argument is reparameterized in terms of angles, letting :

The first few polynomials, parameterized this way, are:

\begin{align} P_0^0(\cos\theta) & = 1 \\ P_1^0(\cos\theta) & = \cos\theta \\ P_1^1(\cos\theta) & = -\sin\theta \\ P_2^0(\cos\theta) & = \tfrac{1}{2} (3\cos^2\theta-1) \\ P_2^1(\cos\theta) & = -3\cos\theta\sin\theta \\ P_2^2(\cos\theta) & = 3\sin^2\theta \\ P_3^0(\cos\theta) & = \tfrac{1}{2} (5\cos^3\theta-3\cos\theta) \\ P_3^1(\cos\theta) & = -\tfrac{3}{2} (5\cos^2\theta-1)\sin\theta \\ P_3^2(\cos\theta) & = 15\cos\theta\sin^2\theta \\ P_3^3(\cos\theta) & = -15\sin^3\theta \\ P_4^0(\cos\theta) & = \tfrac{1}{8} (35\cos^4\theta-30\cos^2\theta+3) \\ P_4^1(\cos\theta) & = - \tfrac{5}{2} (7\cos^3\theta-3\cos\theta)\sin\theta \\ P_4^2(\cos\theta) & = \tfrac{15}{2} (7\cos^2\theta-1)\sin^2\theta \\ P_4^3(\cos\theta) & = -105\cos\theta\sin^3\theta \\ P_4^4(\cos\theta) & = 105\sin^4\theta \end{align}

For fixed m, are orthogonal, parameterized by θ over, with weight :

Also, for fixed ℓ:

In terms of θ, are solutions of

More precisely, given an integer m0, the above equation has nonsingular solutions only when for ℓ an integer ≥ m, and those solutions are proportional to .

Read more about this topic:  Associated Legendre Polynomials

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