Questions tagged [legendre-polynomials]
For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.
2
votes
1answer
72 views
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?
0
votes
1answer
27 views
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 $k \in \mathbb{N}, k<s$.
0
votes
1answer
20 views
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 ...
1
vote
1answer
37 views
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^...
0
votes
1answer
26 views
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(...
5
votes
1answer
57 views
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 ...
1
vote
0answers
17 views
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 ...
0
votes
0answers
19 views
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{...
0
votes
1answer
34 views
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 ...
0
votes
0answers
14 views
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{(\...
2
votes
0answers
35 views
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 ...
3
votes
1answer
60 views
$L^2$ 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 ...
0
votes
0answers
22 views
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}
\...
10
votes
0answers
361 views
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 ...
1
vote
0answers
61 views
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 ...
0
votes
1answer
38 views
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 ...
0
votes
0answers
49 views
Integrals involving Legendre polynomials and exponentials
I found the following integrals in the article here:
$$I_1(x,\lambda)=\int_{-1}^{1}P_n(x')\,\mathrm{e}^{-(x-x')\,\lambda}\,\mathrm{d}x=\mathrm{e}^{-\lambda\,x}\sum_{k=0}^n\frac{(n+k)!\,(-1)^n}{\...
1
vote
1answer
60 views
How to derive closed form expression for the zeros of Legendre's Polynomials (Gaussian-Quadrature formulas)?
In Gaussian quadrature formula of integration we need to have the zeros of Legendre's polynomials. Although we may find the zeros numerically, I got closed form formulas such as $$\pm\frac{1}{3}\sqrt{...
1
vote
1answer
51 views
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 ...
0
votes
0answers
24 views
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.
$P_0^m = 1$
$P_1^m = x$
$P_2^m = \frac{3}{2}x^2-\frac{1}{2}$...
0
votes
1answer
43 views
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)&...
0
votes
1answer
34 views
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.
...
0
votes
0answers
17 views
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=\...
0
votes
2answers
72 views
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 $$u^j=\sum_{q=0}^{j}b_{qj}\phi_q(u), ...
1
vote
1answer
36 views
integral of sum of function involving legendre polynomials
Consider the integral $$\int_{-1}^1\left(\sum_{j=1}^n\sqrt{j(2j+1)}P_j(x)\right)^2dx$$
How to evaluate the integral? Specifically, say, if $n=5$, then what would be the value? Which property of ...
1
vote
1answer
53 views
Why does $\int_{-1}^1 ((1-x^2)(P_m'P_n-P_n'P_m))'\,dx = 0$ for Legendre polynomials?
I was looking at a proof of the orthogonality of the Legendre polynomials in Lebedev's Special Functions and their Applications:
I can't understand why the integral of the first term vanishes. I ...
0
votes
0answers
10 views
legendre integration with derivative terms
I want to prove that integration from -1 to 1 (1-x^2)p'n(x)p'n(x)dx = 2n(n+1)/(2n+1)
I substituted some recurrence relations but got stuck at evaluating the integral of x^2p'p .. how can i deal with ...
0
votes
2answers
33 views
Legendre - orthogonality related proof
How can I proceed to prove that there are constants $α_0, α_1, ..., α_n$ such that
$x^n = α_0P_0(x) + α_1P_1(x) + ... + α_nP_n(x)$ where $P_n$ is legendre polynomial.
I guess that this has to do with ...
1
vote
1answer
36 views
On Legendre's Polynomial
I want to show that the coefficient of $x^n$ in $P_n(x)$ is $(2n)!/ (2^n(n!)^2)$
my problem is that I cannot find the the $n$-th derivative of $(x^2-1)^n$ to be able to simplify Rodrigues' formula of ...
0
votes
0answers
45 views
Integral of Legendre polynomials and Associated Legendre polynomials
I am trying to find the solution to the following integrals. I am quite sure these are well known, but somehow i did not find them already solved online:
1) $\int_0^\pi P_l(\cos{\theta}) P_{l'}^{\pm ...
0
votes
1answer
50 views
Series Solution to Legendre Equation
My professor taught us the series solution to ODE's method today in class, and one of our homework problems was to solve the Legendre Equation.
$$\text{Legendre Equation:} \frac{d^2y}{dx^2}(1-x^2) -...
0
votes
1answer
63 views
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
...
0
votes
1answer
54 views
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 ...
0
votes
0answers
72 views
Sum and Integration with Legendre Polynomial
What is the value of the following infinite sum after integrating the product of two Legendre Polynomials $P_m^0,~P_n^1$, $$\sum_{n=1}^{\infty}\sum_{m=0}^{\infty} \frac{1}{\sqrt{n(n+1)}} A_m\, A_n \...
0
votes
0answers
59 views
Solving the following linear ODE by a numerical method
ODE:
$$y'(x)+3y(x)=1$$
Initial condition: $y(0)=0$
We know that the exact solution is:
$y \left( t \right) =1/3-1/3\,{{\rm e}^{-3\,t}}.$
My Objective:
I want to solve the ODE by Legendre wavelets ...
0
votes
0answers
35 views
The integration of Legendre functions
We know the integration of Legendre wavelet function is $\int_{0}^{T}\Psi(s)ds=P.\Psi(t)$. We can find the matrix $P$ as follows.
My question: I want to learn how to find Matrix $P$. I can' t ...
1
vote
0answers
80 views
Solving 2nd order ODE with variable coefficients
ODE:
$$X''(t)+A(t)X'(t)+B(t)X(t)=F(t),$$
IC's: $$X(0)=U_1, $$
$$X'(0)=U_2$$
where
$X(t)=[X_1 X_2...X_n]^T$ and $U_1, U_2,F(t)$ are $n\times1$ matrices,
$A(t), B(t)$ are $n\times n$ matrices.
...
5
votes
2answers
209 views
Newtonian potential expansion identity
Preliminaries
Consider the Newtonian potential $$\frac{1}{|\vec x - \vec y|}$$ with $\vec{x}, \vec{y} \in \mathbb{R}^3$ and $|\vec{x}| = x > y = |\vec{y}|$. Its Taylor expansion is given by $$\...
2
votes
1answer
110 views
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 ...
0
votes
1answer
55 views
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^...
1
vote
0answers
90 views
Use Rodrigues’formula to generate the Legendre Polynomials
may any one tell how the circled step was given
1
vote
1answer
56 views
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 \...
-3
votes
1answer
126 views
Legendre's Equation, sturm liouville - eigenvalues/eigenfunction
Linear Differential Equation,Legendre's Equation, sturm liouville - eigenvalues/eigenfunction
Consider the linear differential operator:
$$ L = \frac{1}{4}(1+x^2)\frac{d^2}{dx^2}+\frac{1}{2}x(1+x^...
0
votes
0answers
61 views
Orthogonality of Legendre polynomials with logarithmic functions
I have to find the value of this integral:
$\int_{-1}^1 \ln(1-x)*P_3(x)\,dx$
where $P_3(x)$ is the Legendre polynomial.
I thought I can write $\ln(1-x)$ as a summation of Legendre polynomials and ...
0
votes
1answer
87 views
proof (Legendre polynomial )
I'm stuck, couldn't figure it out. I appreciate your help.
Show that :
$$\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)}$$
1
vote
1answer
52 views
Series Legendre Polynomial
We have to prove
$$\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}{...
0
votes
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 $...
0
votes
0answers
25 views
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 ...
0
votes
0answers
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 ...
1
vote
1answer
73 views
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} ...
