Mathematics

Questions tagged [legendre-polynomials]

For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.

The Legendre polynomials $P_n$ are defined to be solutions of Legendre's differential equation

$$\frac{d}{dx} \left[(1 - x^2) \frac{d}{dx} P_n(x)\right] + n(n + 1) P_n(x) = 0$$

Alternatively, by Rodrigues' formula,

$$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \left[(x^2 - 1)^n\right]$$

The Legendre polynomials occur frequently in physics, and in particular in solving Laplace's equation in spherical coordinates.

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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 $1$)? Except for low-degree cases, the roots can't be…
user3180
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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\pi^2}\,\Gamma\left(\frac{1}{4}\right)^4 $ is:…
Jack D'Aurizio
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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…
Chris
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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 I have to use Legendre symbol from the answer. I…
Guillermo
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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 formulas for the same, but i didn't get any intuition…
supermeat boy
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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: $$2\sum_{m\geq 0}\frac{4m+3}{4m+4}\cdot\frac{(-1)^m}{4^m}\binom{2m}{m}\cdot P_{2m+1}(x)\tag{1}$$ with $P_n(x)$ being…
Jack D'Aurizio
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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, and I am curious about the existence of this…
Setness Ramesory
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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 integration one may prove…
Kemono Chen
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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 should be possible to use the substitution…
Grautus
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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
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$I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$

Is there a closed form integral for $$I(x) = -\int_0^1 \frac{1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+ z^2}}{2}\right)\,dz$$ for $-1 < x < 1$ ? This integral is related to Legendre polynomials as $$\frac{-1}{z}\ln\left(\frac{1-x z + \sqrt{1-2 x z+…
bkocsis
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Integral Expression of Legendre Polynomials

a) Verify that for $x > 1$ , $n \in \mathbb{N}$ the function $$P_n(x) = \frac{1}{\pi} \int_0 ^ \pi (x + \sqrt{x ^ 2- 1} \cos \phi) ^ n d \phi$$ is a polynomial of degree $n$ (the $n$ th Legendre polynomial). b) Show that $$P_n(x) = \frac{1}{\pi}…
Ziyang Guo
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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 polynomial exists. In the end there is a "explanation"…
Pinteco
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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: $$\color{blue}{\displaystyle\int_{x=-1}^{1}[{P_{L}}^m(x)]^2\,\mathrm{d}x=\left(\frac{2}{2L+1}\right)\frac{(L+m)!}{(L-m)!}}$$…
BLAZE
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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 $\frac{\log (1-x)}{x}=\sum _{n=0}^{\infty } 2 (-1)^{n-1} (2 n+1) P_n(2 x-1) \left(\sum _{k=n+1}^{\infty } \frac{(-1)^{k-1}}{k^2}\right)$ Can be used to…
Infiniticism
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