# Questions tagged [legendre-functions]

This tag is for questions relating to Legendre Functions (or Legendre Polynomials), solutions of Legendre's differential equation (generalized or not) with non-integer parameters.

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### Legendre Polynomials proving a relation using Rodrigue's Formula

I have been trying to prove the relation$\left (P'_n(1)=\frac{1}{2}n(n+1)\right )$ and have proved it directly from the Legendre differential equation as $P_n(x)${the Legendre polynomial} is the ...
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### Does Associated Legendre Function of Second Kind Give Delta Function?

Associated Legendre Function of Second Kind is singular at $x=\pm 1$. So I am wondering whether it satisfies the corresponding differential equation everywhere or there is a hidden functional of delta ...
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### derivative of normalized associated Legendre function at the limits of x = +/-1

I'm trying to compute the derivative of the normalized associated Legendre function of $x=\cos \theta$ and I'm having issues when $\cos \theta =\pm 1$ which causes the denominator to go to $0$. I've ...
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### find $a_L$ in $\frac{Y_{\ell}^{m}Y_{\ell'}^{-m}}{\sin^2\theta}=\sum_{L=0}^{\ell+\ell'}a_LY_L^0$

Can you prove that $$\frac{Y_{\ell}^{m}Y_{\ell'}^{-m}}{\sin^2\theta}=\sum_{L=0}^{\ell+\ell'}a_LY_L^0~,$$ where $Y_{\ell}^{m}$ denotes the spherical harmonics of degree $\ell$ and order $m$ (both are ...
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### How to convert a hypergeom function to the Legendre function?

Anyone can help me to convert the following maple pdsolve expressed by the hypergeom function to the $LegendreP(n,b,x)$ or $Q$ function? dsolve\Big( (1-x^2)\cdot \frac{d^2 y(x)}{dx^...
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### Equivalence of spherical harmonic definitions

In Griffith's "Introduction to quantum mechanics", the spherical harmonics are defined for integers $l$ and $-l\leq m\leq l$ as Y_{l}^m(\theta,\phi) = \epsilon \sqrt{\frac{2l+1}{4\pi}\frac{(l-|m|)!}...
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