Associated Legendre Polynomial

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Associated Legendre Polynomial:

$$P_{\ell }^m(x)=(-1{)}^m(1-x^2{)}^{m/2}\frac{d^m}{d{x}^m}{P}_{\ell }(x)$$

 

이때 $x=\cos \theta$라 한다면

$$P_{\ell }^m(x=\cos \theta )=(-1{)}^m(1-{\cos }^2\theta {)}^{m/2}\frac{d^m}{d(\cos \theta {)}^m}{P}_{\ell }(x=\cos \theta )$$

 

$$P_{\ell }^m(\cos \theta )=(-1{)}^m({\sin }^2\theta {)}^{m/2}\frac{d^m}{d(\cos \theta {)}^m}{P}_{\ell }(\cos \theta )$$

$$\frac{dx}{d\theta }=-\sin \theta$$

$$dx=-\sin \theta d\theta$$

$$d{x}^m=(-\sin \theta {)}^{m}d{\theta }^m$$

$$P_{\ell }^m(\cos \theta )=(-1{)}^m(\sin \theta {)}^m\frac{d^m}{(-1{)}^m{\sin }^m\theta d{\theta }^m}{P}_{\ell }(\cos \theta )$$

 

$$P_{\ell }^m(\cos \theta )=\frac{d^m}{d{\theta }^m}{P}_{\ell }(\cos \theta )$$

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Reference

Wikipedia, Associated Legendre polynomials