Questions tagged [legendre-polynomials]

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

386 questions
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Let $p$ be a prime of the form $p = a^2 + b^2$ with $a,b \in \mathbb{Z}$ and $a$ an odd prime. Prove that $(a/p) =1$

Let $p$ be a prime of the form $p = a^2 + b^2$ with $a,b \in \mathbb{Z}$ and $a$ an odd prime. Prove that $(a/p) =1$ Could anyone give me a hint for the solution please?
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Rodrigues' formula proof [closed]

Consider the polynomials $P_s (x) = \frac{1}{2^ss!}\frac{d^s}{dx^s}[(x^2-1)^s]$. Prove $\int_{-1}^{1}x^kP_s(x)dx = 0$ for all $k \in \mathbb{N}, k<s$.
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Integral with Legendre Polynomial $\int x^2(P_l(x))^2dx$

I have the integral where $P_l$ is the $l$th legendre polynomial. $$\int^1_{-1}x^2\left(P_l(x)\right)^2dx$$ I think the way to do this is by integration by parts but I am not sure how to start. I ...
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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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Expressing spherical functions with zonal harmonics

$\newcommand{\d}{\mathrm d}$First time using StackExchange, so please excuse any mistakes: My goal is to express functions on the sphere as a sum of zonal harmonics. I've spent a while with the ...
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$L^2$ norm of the first derivative of Legendre polynomials

Recently I have encountered a question while studying the orthogonal properties of Legendre's polynomial $$\int_{-1}^{1} P_{n}^{\prime}(x) P_{n}^{\prime}(x)=n(n+1), n\geq1$$ I have tried the ...
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Proof for a Legendre polynomial identity on a 2-sphere that appears in cosmology

How can I prove the following identity? \begin{equation} \int d\Omega_{\hat k} P_l(\hat n \cdot \hat k) P_{l'}(\hat n'\cdot \hat k)=\frac{4\pi}{2l+1} P_l(\hat n\cdot \hat n')\delta_{ll'}\label{PP} \...
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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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Prove that the Legendre polynomial holds true using the generating function and a binomial expansion.

I tried using that $$P_{2n}(z)=\frac1{2^{2n}(2n)!}\frac{d^{2n}}{dz^{2n}}(z^2-1)^{2n}$$ but I am having trouble taking the derivative when $n$ is unknown and $z$ is $0$. Any help would be greatly ...
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Orthogonal Property of Legendre Polynomials

How can I get $$nu_{n} + (n-1)u_{n-1},$$ where $$u_{n} = \int_{-1}^{1} x^{-1}P_{n-1}(x) P_{n}(x)\, \mathrm {d}x\; ?$$ I did many search, and also I did try by myself. But without a success. Is ...
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Domain limitations on generating function for Legendre polynomials

The generating function for legendre polynomials is $$\frac{1}{\sqrt(1-2ut+u^2)}=\sum_{i=0}^\infty u^iP_i(t)$$ where $P_i$ is $i^{th}$ legendre polynomial however for binomial expansion of left hand ...
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Confirm that P_l^m satisfy associated Legendre

How do I confirm that $P_l^m$ for $l = 0, 1, 2$ satisfy the associated Legendre with eigen-values $l(l+1)$? The polynamials should be. $P_0^m = 1$ $P_1^m = x$ $P_2^m = \frac{3}{2}x^2-\frac{1}{2}$...
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Orthogonality of the First Four Legendre Polynomials

Using the recurrence relation $$(n+1)P_{n+1}=x(2n+1)P_n(x)-nP_{n-1}(x) \ \ n\geq 1,$$ I've calculated the first four Legendre Polynomials as \begin{align} P_0(x)&=1 \\ P_1(x)&=x \\ P_2(x)&...
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Why the zeros of the orthogonal polynomials are symmetric about the origin if the weight function is even?

Why the zeros of the orthogonal polynomials are symmetric about the origin if the weight function is even? t's killing me and every book i've seen is left as an exercise. I'm studying Gauss Legendre. ...
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Use Rodrigues’formula to generate the Legendre Polynomials

may any one tell how the circled step was given
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Equivalence of norms on $\mathbb{P}^n$ space of polynomials of degree at most $n$

I'm stuck on a problem and just need hints to keep going. Thanks. Let $p_k\in \mathbb{P}^k$ be the degree $k$ real valued Legendre polynomial on $[0,1]$. These form an orthonormal basis wrt the inner ...
verification of my solution to $((1-x^2)y')'=\sum_{n=1}\tfrac{P_n(x)}{2^n}$ and possibly suggestions on how to solve the resulting integral
Given that : $$((1-x^2)y')'=\tfrac{2}{(5-4x)^{1/2}}-1$$ and that $$\tfrac{2}{(5-4x)^{1/2}}=\sum_{n=0}\tfrac{P_n(x)}{2^n}$$ we have that $$((1-x^2)y')'=\sum_{n=1}\tfrac{P_n(x)}{2^n}$$ We want ...
I am currently working on an assignment with Legendre Polynomials. The integral I get stuck with is in fact the integral of the Legendre Polynomial itself i.e. \int \frac{1}{2^n n!} \frac{d^n}{dt^n} ...