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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Roots of Legendre Polynomial
I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials.
Are the roots always simple (i.e., multiplicity $1$)?
...
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25
votes
3
answers
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An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$
When playing with the complete elliptic integral of the first kind and its Fourier-Legendre expansion, I discovered that a consequence of $\sum_{n\geq 0}\binom{2n}{n}^2\frac{1}{16^n(4n+1)}=\frac{1}{16\...
- 355k
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votes
0
answers
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How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials
By experimenting in Mathematica, I have found the following expression for the integral:
$$
\int_{b-a}^{b+a}\sigma h_{n}^{(1)}(\sigma)P_{n}^{m}\left(\frac{\sigma^{2}-a^{2}+b^{2}}{2b\sigma}\right)P_{n'}...
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The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $ p-\left(\frac{-ab}{p}\right)$
What I need to show is that
For $\gcd(ab,p)=1$ and p is a prime, the number of solutions of the equation $ax^2+by^2\equiv 1\pmod{p}$ is exactly $$p-\left(\frac{-ab}{p}\right)\,.$$
I got a hint that ...
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13
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1
answer
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Intuition of orthogonal polynomials?
Recently, while learning about the regularization methodologies(in machine learning), i came across orthogonal polynomials ( of Legendre's polynomials), I looked up on the Internet and there are ...
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12
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1
answer
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About the Gibbs phenomenon for the Legendre polynomials as an orthogonal base of $L^2(-1,1)$
In a recent question, I proved that the Fourier-Legendre expansion of the function $f(x)=\text{sign}(x)$ over $(-1,1)$ is given by:
$$2\sum_{m\geq 0}\frac{4m+3}{4m+4}\cdot\frac{(-1)^m}{4^m}\binom{2m}{...
- 355k
10
votes
1
answer
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Legendre Polynomial Orthogonality Integral
I want to show that $$\int_{-1}^{1}L_n(x)L_m(x)dx$$ is zero for $m<n$ and $\frac{2}{2n+1}$ for $m=n$. For $m<n$, I want to apply $(x^2-1)^n=(x-1)^n(x+1)^n$ and integration by parts. For $m=n$ it ...
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Evaluate $\int_0^1 x^{a-1}(1-x)^{b-1}\operatorname{Li}_3(x) \, dx$
Define
$\small f(a,b)=\frac1{B(a,b)}\int_0^1 x^{a-1}(1-x)^{b-1} \text{Li}_3(x) \, dx$$ $$=\frac a{a+b}{}_5F_4(1,1,1,1,a+1;2,2,2,1+a+b;1)$
Where $a>-1$ and $b>0$.
$1$. By using contour ...
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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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Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$
The question that follows is needed as part of a derivation of the Associated Legendre Functions Normalization Formula: $$\color{blue}{\displaystyle\int_{x=-1}^{1}[{P_{L}}^m(x)]^2\,\mathrm{d}x=\left(\...
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Trying to prove that $\pi$ is irrational using Legendre Polynomials.
Unfortunately, numerical data sugggest that is not possible to show that $\pi$ is irrational with the polynomials below. I've to search for another polynomial... But I've little faith that such ...
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8
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Proof that Legendre Polynomials are Complete
Can somebody either point me to, or show me a proof, that the Legendre polynomials, or any set of eigenfunctions, are complete?
anon
8
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1
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Fourier Legendre expansion for $\frac{\text{Li}_2(x)}{x}$
Background: I'm computing harmonic series using FL expansion. For instance, the following
$\frac{\log (1-x)}{x}=\sum _{n=0}^{\infty } 2 (-1)^{n-1} (2 n+1) P_n(2 x-1) \left(\sum _{k=n+1}^{\infty } \...
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votes
1
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Integral involving Legendre polynomial and $x^n$
I am trying to show that,
\begin{align}
I = \int_{-1}^1 x^nP_n(x)\,\mathrm{d}x = \frac{2^{n+1}n!n!}{(2n+1)!}
\end{align}
So far I have done the following. Rodrigues formula is as follows:
\begin{...
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votes
2
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Why does the DLMF distinguish between Ferrers functions and the associated Legendre functions?
In the introduction to the chapter on Legendre functions, the DLMF starts off with the following notations
The main functions treated in this chapter are the Legendre functions $\mathsf{P}_\nu(x)$,...
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