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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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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Differing Forms of Legendre Polynomials

Almost all sources of information quoting the Legendre Polynomials includes the following forms (I have included the first 4 polynomials): $P_0(x) = 1 $ $P_1(x) = x $ $P_2(x) = 1/2(3x^2-1) $ $P_3(...
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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 ...
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Numerically approximating functions on the sphere by rotating zonal harmonics.

This question extends the approach mentioned in Expressing spherical functions with zonal harmonics, which I will briefly restate for ease of reading: I would like to express any function on the ...
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Legendre Polynomial of Second Kind-Neumann's Formula

In textbook Mathews&Walkers problem 7.6 Starting from \begin{equation*} Q_n(z)=\frac{1}{2} P_n(z)\ln\left( \frac{z+1}{z-1}\right)+f_{n-1}(z) \end{equation*} we can derive Neumann's Formula \begin{...
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Recurrence relations and power series solution

I am given the following initial value problem: $$(1-x^2)y''+7xy'-26y=0 \qquad , \qquad y(0)=0 \qquad , \qquad y'(0)=4$$ I have solved for the singular points, which are $x= 1, -1$ The question ...
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Legendre polynomial expansion of a positive function

Let´s assume I have a function $f(\theta)>0$ defined for $\theta<\pi$ and $\theta >0$. I want to find its Legendre polynomial decomposition $f(\theta)= \sum_{l=0}^\infty f_l \, P_l(\cos{(\...
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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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1answer
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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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Integrals involving Legendre polynomials and exponentials

I found the following integrals in the article here: $$I_1(x,\lambda)=\int_{-1}^{1}P_n(x')\,\mathrm{e}^{-(x-x')\,\lambda}\,\mathrm{d}x=\mathrm{e}^{-\lambda\,x}\sum_{k=0}^n\frac{(n+k)!\,(-1)^n}{\...
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How to derive closed form expression for the zeros of Legendre's Polynomials (Gaussian-Quadrature formulas)?

In Gaussian quadrature formula of integration we need to have the zeros of Legendre's polynomials. Although we may find the zeros numerically, I got closed form formulas such as $$\pm\frac{1}{3}\sqrt{...
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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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Integrate orthogonal function over solid angle

How do I integrate a product of Legendre polynomials over a volume? So I understand that bunch of complete basis orthogonal basis as well. i.e. for Legendre polynomial, $\int_{-1}^1P_n(x)P_m(x)dx=\...
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Polynomial identity [Solved]

In the book I am studying, the author says: Since $\phi_q$ is a polynomial of degree $q$, for all $j=1,2, \dots, l$, there exist real numbers $b_{qj}$ such that $$u^j=\sum_{q=0}^{j}b_{qj}\phi_q(u), ...
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integral of sum of function involving legendre polynomials

Consider the integral $$\int_{-1}^1\left(\sum_{j=1}^n\sqrt{j(2j+1)}P_j(x)\right)^2dx$$ How to evaluate the integral? Specifically, say, if $n=5$, then what would be the value? Which property of ...
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Why does $\int_{-1}^1 ((1-x^2)(P_m'P_n-P_n'P_m))'\,dx = 0$ for Legendre polynomials?

I was looking at a proof of the orthogonality of the Legendre polynomials in Lebedev's Special Functions and their Applications: I can't understand why the integral of the first term vanishes. I ...
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legendre integration with derivative terms

I want to prove that integration from -1 to 1 (1-x^2)p'n(x)p'n(x)dx = 2n(n+1)/(2n+1) I substituted some recurrence relations but got stuck at evaluating the integral of x^2p'p .. how can i deal with ...
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Legendre - orthogonality related proof

How can I proceed to prove that there are constants $α_0, α_1, ..., α_n$ such that $x^n = α_0P_0(x) + α_1P_1(x) + ... + α_nP_n(x)$ where $P_n$ is legendre polynomial. I guess that this has to do with ...
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On Legendre's Polynomial

I want to show that the coefficient of $x^n$ in $P_n(x)$ is $(2n)!/ (2^n(n!)^2)$ my problem is that I cannot find the the $n$-th derivative of $(x^2-1)^n$ to be able to simplify Rodrigues' formula of ...
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Integral of Legendre polynomials and Associated Legendre polynomials

I am trying to find the solution to the following integrals. I am quite sure these are well known, but somehow i did not find them already solved online: 1) $\int_0^\pi P_l(\cos{\theta}) P_{l'}^{\pm ...
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Series Solution to Legendre Equation

My professor taught us the series solution to ODE's method today in class, and one of our homework problems was to solve the Legendre Equation. $$\text{Legendre Equation:} \frac{d^2y}{dx^2}(1-x^2) -...
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Differentiate Legendre’s equation $m$ times using Leibniz' rule for differentiating products

$$(1-x^2)u''(x)-2xu'(x) + \ell(\ell+1)u(x)=0\tag{1}$$ Assume that $m$ is non-negative, differentiate $(1)$ (Legendre’s equation) $m$ times using Leibniz' theorem for differentiation to show that ...
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How to tell if series terminates (Legendre ODE)

when solving for the coefficients for the Legendre ODE $(1-x^2)y’’-2xy’+l(l+1)y=0$, I understand how to obtain the recurrence relation $$a_{k+2}=\frac{k(k+1)-l(l+1)}{(k+2)(k+1)}a_k.$$ What I do not ...
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Sum and Integration with Legendre Polynomial

What is the value of the following infinite sum after integrating the product of two Legendre Polynomials $P_m^0,~P_n^1$, $$\sum_{n=1}^{\infty}\sum_{m=0}^{\infty} \frac{1}{\sqrt{n(n+1)}} A_m\, A_n \...
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Solving the following linear ODE by a numerical method

ODE: $$y'(x)+3y(x)=1$$ Initial condition: $y(0)=0$ We know that the exact solution is: $y \left( t \right) =1/3-1/3\,{{\rm e}^{-3\,t}}.$ My Objective: I want to solve the ODE by Legendre wavelets ...
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The integration of Legendre functions

We know the integration of Legendre wavelet function is $\int_{0}^{T}\Psi(s)ds=P.\Psi(t)$. We can find the matrix $P$ as follows. My question: I want to learn how to find Matrix $P$. I can' t ...
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Solving 2nd order ODE with variable coefficients

ODE: $$X''(t)+A(t)X'(t)+B(t)X(t)=F(t),$$ IC's: $$X(0)=U_1, $$ $$X'(0)=U_2$$ where $X(t)=[X_1 X_2...X_n]^T$ and $U_1, U_2,F(t)$ are $n\times1$ matrices, $A(t), B(t)$ are $n\times n$ matrices. ...
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Newtonian potential expansion identity

Preliminaries Consider the Newtonian potential $$\frac{1}{|\vec x - \vec y|}$$ with $\vec{x}, \vec{y} \in \mathbb{R}^3$ and $|\vec{x}| = x > y = |\vec{y}|$. Its Taylor expansion is given by $$\...
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1answer
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Using Frobenius method to solve the Legendre differential equation

I'm tasked with solving the Legendre differential equation, and Using $c=0$, obtain a series of even powers of $x$ (with $a_1=0$). I found this exercise to be good at highlighting what I found ...
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Integrating multiple products of order 1 Legendre functions

I would like a closed form expression for the integrals of some products like the following: $$ \int_{-1}^1 P^1_j P^1_k P^1_l P^1_m dx $$ and $$ \int_{-1}^1 P^0_j P^0_k P^1_l P^1_m dx, $$ where $P^...
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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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Find the Legendre polynomial

Let us consider the numerical integral $ \ \int_{-1}^{1}w(x) f(x)dx=\sum_{i=0}^{N} f(x_i)w_i$, where $w_i$ are the weights and $w(x)$ is the weight function. Legendre polynomials, denoted by $ \{p_n \...
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Legendre's Equation, sturm liouville - eigenvalues/eigenfunction

Linear Differential Equation,Legendre's Equation, sturm liouville - eigenvalues/eigenfunction Consider the linear differential operator: $$ L = \frac{1}{4}(1+x^2)\frac{d^2}{dx^2}+\frac{1}{2}x(1+x^...
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Orthogonality of Legendre polynomials with logarithmic functions

I have to find the value of this integral: $\int_{-1}^1 \ln(1-x)*P_3(x)\,dx$ where $P_3(x)$ is the Legendre polynomial. I thought I can write $\ln(1-x)$ as a summation of Legendre polynomials and ...
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1answer
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proof (Legendre polynomial )

I'm stuck, couldn't figure it out. I appreciate your help. Show that : $$\frac{1}{2} \ln \left |\frac{1+x}{1-x} \right |=\sum_{n \text { odd}} \frac{(2n+1)}{n(n+1)}\mathcal {P_n(x)}$$
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1answer
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Series Legendre Polynomial

We have to prove $$\ln \Big(\frac{x+1}{1-x}\Big)=\sum_{n≥0} \frac{x^{n+1}}{n+1}P_n(x)$$ using the generatrix function of Legendre polynoms. I don't know if it is useful, but $$\int_{-1}^{1}\frac{1}{...
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0answers
21 views

Fourier-Legendre Series for $\arcsin{x}$

I have been trying to work out how to work out the coefficients in the Fourier-Legendre series of $\arcsin{x}$ (i.e., find $c_n$'s s.t. $\arcsin{x}=\sum_{n=0}^\infty c_n P_n(x)$ where $P_n(x)$ is the $...
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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 ...
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51 views

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 ...
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1answer
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Can we express the integral of the nth derivative of this function analytically?

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} ...