Associated Legendre Polynomial Notes| A Solution to The Problems | Mathematical Physics
Associated Legendre Polynomial Notes| A Solution to The Problems
According to Wikipedia's definition
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation
![{\displaystyle \left(1-x^{2}\right){\frac {d^{2}}{dx^{2}}}P_{\ell }^{m}(x)-2x{\frac {d}{dx}}P_{\ell }^{m}(x)+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right]P_{\ell }^{m}(x)=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/914d8bc4e6fd78db7db32609c345663927ad11ab)
or equivalently
![{\displaystyle {\frac {d}{dx}}\left[\left(1-x^{2}\right){\frac {d}{dx}}P_{\ell }^{m}(x)\right]+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right]P_{\ell }^{m}(x)=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8c86f72c4096479b00d25aaa318916c46e36c9)
where ℓ the degree of the associated Legendre polynomial
and m is the order of the associated Legendre polynomial.
This equation has nonzero solutions that are nonsingular on [−1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values.
In the attached document you'd find a solution to most important questions relating to associated Legendre polynomial.
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