Questions tagged [legendre-polynomials]
For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.
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Legendre addition theorem in $2$ dimensions
We know the addition theorem for Legendre polynomials in spherical coordinates is
$$P_\ell(\cos\gamma)=\dfrac{4\pi}{2\ell +1}\sum_{m=-\ell}^\ell\mathrm Y_{\ell m}(\theta_1,\phi_1)\,\mathrm Y_{\ell m}^\...
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Proof involving integrals, binomial coefficients and Legendre polynomials
I'm currently working on a research paper involving L-moments statistics. This is the first time for me working on that subject.
I stumbled upon this article mentioning a very important equality, ...
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Derivation of Legendre Polynomials from only orthogonality
I recently stumbled on the idea of Legendre polynomials, from the perspective of using them to approximate functions over a region, and I discovered all of these other ways to express them, using ...
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Deriving quadrature weights from discrete orthogonality of exponentials
In the proof of Lemma 2 of Driscoll and Healy, it says
\begin{align}
\sqrt{2}\delta_{k,0} &= \frac{1}{\sqrt{2}}\int_0^\pi P_k(\cos\theta)\sin\theta d\theta\\\\
&= \frac{1}{2\sqrt{2}}\int_{-\pi}...
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Legendre's equation of order n in differential equations
The equation
$$(1-x^2)y'' - 2xy'+n(n+1)y = 0$$
is called Legendre's equation of order $n$.
I need to show that this equation of order $1$ has $y=x$ as one solution and then I need to use this to ...
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Could someone explain the reason behind using Legendre Polynomials?
Where $x = \cos(\phi)$ and therefore (and this is very important) $x=[-1,1]$, we have the (special such that m = 0) associated Legendre Equation $(1-x^2)\frac{d^2 y}{dx^2} - 2x \frac{dy}{dx}+ky = 0$, ...
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Derivation of the associated Legendre Polynomials
I have been struggling to find a proper derivation of the associated Legendre Polynomials and a derivation of
$$P_l^{-m}(\mu)=(-1)^m\frac{(l-m)!}{(l+m)!}P_l^m(\mu)$$
Can someone point to a proper ...
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How to find an expression for the $n$th partial derivatives of $1/r$?
From Pirani's lectures on General Relativity I got the following identity which he asks the reader to prove by induction which is easy
$$\partial_{\alpha_1} \partial_{\alpha_2} ... \partial_{\alpha_n} ...
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Clarification of legendre polynomials from WolframMathWorld
I don't understand the way they got to the angle-based equation. What I did is in blue, what seems to be the problem in pink.
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Simplify in closed-form $\sum_n P_n(0)^2 r^n P_n(\cos \theta)$
Simplify in a closed form the sum $$S(r,\theta)=\sum_{n=0}^{\infty} P_n(0)^2 r^n P_n(\cos \theta)$$ where $P_n(x)$ is the Legendre polynomial and $0<r<1$. Note that $P_n(0)= 0$ for odd $n$ and $...
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Evaluate $\int_{0}^{1} \operatorname{Li}_3\left [ \left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ] \text{d}x$
Possibly evaluate the integral?
$$
\int_{0}^{1} \operatorname{Li}_3\left [
\left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ]
\text{d}x.
$$
I came across this when playing with Legendre polynomials, ...
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Expansion in terms of legendre polynomial
Obtain the first three terms in the expansion of function in terms of legendre polynomial
F(x) in a series of the form
$$
F(x) = \sum_{k=0}^{\infty} A_k P_k(x)
$$
where
$$F(x)=\{\cos(x) \...
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Proving that the Legendre differential equation has solution of degree $n$ when $p=n$
For $p\geq 0$ the Legendre differential equation is
$$(1-t^2)y''-2ty'+p(p+1)y=0.$$
Two linearly independent solutions that I have found for this diff equation are
$$y_1(t)=1+\sum_{n=1}^\infty(-1)^n\...
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Integration of Legendre polynomials with their derivatives
I need the results of $\int_{-1}^1 p_i(x) p_j'(x) dx$, where $p_i$ is the classical Legendre polynomials in $[-1,1]$. Using the matlab, I get the following result:
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How to prove the following properties about legendre function?
My teacher wrote the following about the legendre function without proof
1) $P^{-m}_{n}(x)=(-1)^{m}\frac{(n-m)!}{(n+m)!}P_{n}^{m}(x)$
2) $P_{2n}(0)=\frac{(-1)^{m}}{\sqrt{\pi}}\frac{\Gamma(n+\frac{1}...
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