Associated Legendre Polynomial

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Associated Legendre Polynomial:

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

 

이때 \(x=\cos\theta\)라 한다면

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

 

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

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

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

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

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

 

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

\(\blacksquare\)

 

Reference

Wikipedia, Associated Legendre polynomials