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Questions tagged [legendre-functions]

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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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Inverse of Legendre Dual

Reading the book "Concentration inequalities A nonasymptotic theory of independence" I came across the following results (Lemma 2.4): Let $f$ be a convex function such that $f(0)=f'(0)=0$ and such ...
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The Overlap Integral of Three Associated Legendre Functions

I obtained the following integral in my (physics) research: $ \int_{-1}^{1} \mathsf{P}_{-\frac{1}{2}+i \mu}^{ik_1}(x) \mathsf{P}_{-\frac{1}{2}+i \mu}^{ik_2}(x) \mathsf{P}_{-\frac{1}{2}+i \mu}^{ik_3}(...
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Transforming an ODE into Legendre's Equation

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{...
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The integration of Legendre functions

We know the integration of Legendre wavelet function is $\int_{0}^{T}\Psi(s)ds=P.\Psi(t)$. We can find the matrix $P$ as follows. My question: I want to learn how to find Matrix $P$. I can' t ...
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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^...
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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 $...
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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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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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Sum of associated Legendre functions

I want to find the sums of two expressions involving the Schmidt-normalized associated Legendre functions. They are defined by \begin{align} S_l^0(x) &= P_l^0(x) \\ S_l^m(x) &= \sqrt{2 \frac{(...
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Legendre Polynomial Of Second Kind

How to calculate the normalisation factors of Legendre Polynomial of second kind? It is provided that ,the normalisation factors are chosen so that second kind Polynomials satisfies the recurrence ...