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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Prove that the Legendre polynomials satisfy $P_n(1) = 1$ and $P_n(-1)=(-1)^n$
Problem. Use the following relationship for the Legendre polynomials
$$ P_n(\cos\theta) = \frac{1}{n!} \frac{\partial^n}{\partial t^n} (1 - 2t\cos\theta + t^2)^{(-1/2)} |_{t=0} $$
in order to prove ...
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Does it make sense to think about formal power series where the coefficients belong to a Ring?
Normally, one defines formal power series as below:
Let $F$ be a field. A formal power series is an expression of the form
$$ a_0 + a_1x + a_2x^2 + \dots = \sum_{n \geqslant 0} a_nx^n,$$
where $\{a_n\}...
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partial alternating sum of legendre polynomial
My question concerns the (generalized) Legendre polynomials in the $d$-dimensional space (see, e.g. Mueller, Freeden & Gutting):
$$
P_n(d;t) = \frac{|S^{d-3}|}{|S^{d-2}|} \int_{-1}^1 (t + i \sqrt{...
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Showing that a series that involves Legendre polynomial converges.
I am currently studying Legendre polynomials and I am trying to prove its generating function specifically using the recurrence relation
$$ (n+1)L_{n+1}(x) = x(2n+1)L_n(x) - nL_{n-1}(x), \quad \forall ...
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Show that $K(u)=\sum_{k=0}^r P_k(0) P_k(u) \mathbf{1}_{\{|u| \leq 1\}}$ is a kernel of order $r$
I have a question concerning the construction of kernels wit orthogonal polynomials. The instructor defined the orthogonal polynomials as
$$P_0(x)=\frac{1}{\sqrt{2}}, P_m(x)=\sqrt{\frac{2 m+1}{2}} \...
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Legendre polynomials: computing $\int_0^1 xP_n(x)\,dx$
I am trying to find the integration:
$$
\int_{0}^{1}{xP_n\left(x\right)}\,dx
$$
I know that I should split Rodrigues's formula up into two parts $P_{2n}\left(x\right)$ for even terms and $P_{2n+1}\...
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further simplification of a summation involving Legendre and Associated Legendre polynomials
Was calculating something from a physics problem and found myself dealing with the following summation:
$$ \sum\limits_{\text{odd} \text{ } l}^{\infty}P_{l+1}(0)P'_l (\cos{\theta})\sin{\theta}\ $$
By ...
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Generating function of orthogonal polynomial basis
I'm studying the bases made up by orthogonal polynomial such as: Hermite, Legendre, Laguerre, Chebyshev. On my book there is a theoretical introduction that gives the difinition of generating function ...
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Integration of a square of Conical (Mehler) function
I want to evaluate
$$\int_{\cos\theta}^1 \left( P_{-1/2+i\tau}(x) \right)^2 dx,$$
where $P$ is the Legendre function of the first kind, $i$ is the imaginary unit, and $\tau$ is a real number.
Are ...
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Answer Check: Best Approximation of $(x+1)e^{-x}$ using Legendre Polynomials
I have worked through and produced an answer for the following question but am unsure of whether or not it is correct. I would appreciate some insight.
$\textbf{Question}$:
Using the Gram-Schmidt ...
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Averaged value of product of Legendre Polynomials
Note: the following question comes from Alex Meiburg via Faceboook and was found via his work with the Legendre Polynomials in quantum machine learning.
Let $P_k$ be the $k$-th Legendre Polynomial.
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Can order of summation and integral be interchanged in : $\int_{-1}^1 ( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime))\text{d}\xi^\prime$?
I am wanting to know if there is a proof that the order of summation and integration can be interchanged in $$\int_{-1}^1 \left( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime)\right)\text{d}\...
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Legendre polynomial as basis for finite element method
When people say use Legendre polynomial as basis polynomial in fem, are they using the polynomials themselves or the integral of those polynomials?
I'm asking this because I have this poisson equation:...
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How to integrate products of Legendre functions over the interval [0,1]
The associated Legendre polynomials are known to be orthogonal in the sense that
$$
\int_{-1}^{1}P_{k}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{k,l}
$$
This is intricately linked to ...
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Series representation of the associated Legendre polynomial
I have found the following identity for the associated Legendre polynomial to be true:
$$
P_{n}^{m}(\tau)=\frac{n!(n+m)!}{2^n}\sum_{s=0}^{n-m}\frac{(-1)^{m+s}(1+\tau)^{n-m/2-s}(1-\tau)^{m/2+s}}{(n-s)!...
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