# 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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How do I express $cos(3\theta)$ and $sin^{2}(\theta)$ in Legendre's polynomials, knowing that $x=cos\theta$? I know that $f(x)=\sum a_{n}P_{n}(x)$ and P_{n}=\frac{(-1)^{n}}{2^{n}n!}\frac{d^{n}}{dx^{n}... 2 votes 1 answer 366 views ### Integrating odd Legendre polynomials using generating function I must show using generating function of Legendre polynomials, that \begin{align} \int_0^1 P_{2n+1}(x)\phantom{1}dx = (-1)^n\frac{(2n)!}{2^{2n+1}n!(n+1)!} \end{align} My attempt is to change the ... 0 votes 0 answers 43 views ### Monotonicity of Legendre functions (of first kind) with respect to theirs order I will denote byP^m_l(x)$the Legendre first kind function of order$m$and of degree$l$,$m$is a positive integer and$l$is real number. I fix the value of$x\in[0,1]$and$l=-0.5$. Is there any ... 0 votes 1 answer 108 views ### How to prove that Spherical Harmonics must have integer order$m$and degree$n$? Spherical harmonics : \begin{equation} Y_{m}^{n}(\theta,\phi) = N_{}\mathcal{P}_{m}^{n}(\cos \phi) e^{in\theta}, \end{equation} are the solution of the two angular equations (Legendre associated eq ... 0 votes 1 answer 244 views ### Convex envelope of a function I am trying to calculate the convex envelope of a function, which is calculated by doing the Legendre transform twice (right?). Therefore, I was trying to calculare the convex envelope of$f(x)=(x^2-1)... 64 views

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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? \begin{equation} dsolve\Big( (1-x^2)\cdot \frac{d^2 y(x)}{dx^...
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I am trying to transform the ODE $$\frac{1}{\sin(\theta)}\frac{d}{d\theta}\left(\sin(\theta)\frac{dS}{d\theta}\right)+\lambda S=0,$$ in to Legendre's equation $$(1-\mu^2)\frac{d^2S}{d\mu^2}-2\mu\frac{... 0 votes 1 answer 245 views ### Integrating multiple products of order 1 Legendre functions I would like a closed form expression for the integrals of some products like the following:$$ \int_{-1}^1 P^1_j P^1_k P^1_l P^1_m dx $$and$$ \int_{-1}^1 P^0_j P^0_k P^1_l P^1_m dx, $$where P^... 3 votes 0 answers 113 views ### An upper bound of the first eigenvalue of Laplacian on a Riemannian manifold. I'm reading the Cheng's thesis ""Eigenvalue Comparison Theorems and Its Geometric Applications," and the author obtains an estimate of eigenvalues of the Laplacian based upon his theorem: If M is ... 1 vote 1 answer 385 views ### Series Legendre Polynomial We have to prove$$\ln \Big(\frac{x+1}{1-x}\Big)=\sum_{n≥0} \frac{x^{n+1}}{n+1}P_n(x)$$using the generatrix function of Legendre polynoms. I don't know if it is useful, but$$\int_{-1}^{1}\frac{1}{...
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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|)!}...