Questions tagged [legendre-polynomials]
For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.
521
questions
5
votes
2answers
83 views
Integral of a product of Legendre polynomials
I would like to show that
$$
\int_{-1}^{1}P_{n}^{1}(x)P_{n'}^{0}(x)\frac{x}{\sqrt{1-x^{2}}}\,\mathrm dx
=
\begin{cases}
-\frac{2n}{2n+1},&n=n'>0\\
-2,&n>n'\text{ and } n-n' \text{ even}\\...
0
votes
0answers
7 views
Convergence of Legendre-Fourier series outside [-1,1]
Good day all,
my numerical computations suggest that common functions, such as $\sin(x)$, $\exp(x)$, when expanded into the Legendre-Fourier series tend to converge also outside the interval [-1,1].
...
-1
votes
0answers
23 views
Integral of first derivatives of associated Legendre polynomials
I would like to calculate the average power which each electromagnetic spherical wave carries. To obtain the power flow, I calculate the Poynting vector and integrate over the surface of a sphere. In ...
2
votes
1answer
58 views
Legendre Polynomials integral
I've been asked to calculate:
$$
\int_{0}^{1} P_{\ell}(x)dx,
$$
where $P_{\ell}(x)$ is a Legendre polynomial by using:
i)The generating function:
$$
\sum_{\ell=0}^{\infty}P_{\ell}(x)t^{\ell}=\frac{1}{\...
0
votes
0answers
13 views
Legendre Coefficients and more for this question
I' struggling to understand these questions, I'm not sure how to find Legendre coefficients set out like this for part a) and d) just confuses me.
I know this isn't a site to ask for help on questions ...
3
votes
0answers
69 views
definite integral solution for $\int^{\pi}_{\alpha} P_{n}(\cos\theta)\:P_{m}(\cos\theta)\sin\theta\:d\theta$
I'm trying to verify the result of $\int^{\pi}_{\alpha} P_{n}(\cos\theta) P_{m}(\cos\theta)\sin\theta\:d\theta$ in a publication which gives:
$$\tfrac{\sin \alpha}{m(m+1) - n(n+1)}\biggl( P_{m}(\cos\...
0
votes
0answers
19 views
Orthogonality of Legendre Polynomials support
I've recently been introduced to orthogonality of Legendre polynomials, I think I understand the idea behind it; that the integral of the products of two polynomials in the range $(1,-1)$ will equal $...
0
votes
1answer
40 views
Orthonormal polynomial basis of $L^2([0,1])$
I was wondering if, given a natural number $i\in \mathbb N$, there exists an orthonormal basis (w.r.t. the standard scalar product) $(p_n)_{n \in \mathbb N}$ of $L^2([0,1])$ such that $p_n$ is a ...
3
votes
1answer
146 views
How to calculate the integral of a Legendre polynomial
I would like to show that
$$
\int_{0}^{1}P_{l}(1-2u^{2})e^{2i\alpha u}du=i\alpha j_{l}(\alpha)h_{l}(\alpha)
$$
where $P_{l}(x)$ are the Legendre polynomials, $\alpha$ is a positive constant and $j_{l}$...
0
votes
1answer
44 views
$\sum_{n=0}^{\infty}\frac{x^{n+1}}{n+1}P_n(x)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$
Using the generating function of Legendre polynomials, show
\begin{equation}
\sum_{n=0}^{\infty}\frac{x^{n+1}}{n+1}P_n(x)=\frac{1}{2}\ln
\left(\frac{1+x}{1-x}\right)
\end{equation}
My attempt
I ...
1
vote
1answer
30 views
Re-arranging and integrating a Generating Function of Legendre Polynomials
I have a generating function of Legendre Polynomials given by: $G(x,r)= \sum_{n=0}^\infty P_n(x)r^n = (1-2rx +r^2)^{-1/2}$
My problem is that I'm asked to find $\int_{-1}^1P_n(x)dx$ but all I have is $...
1
vote
0answers
27 views
Expanding function on [0,1] using Legendre polynomials
A quick question. Given a boundary condition that a function $f(x)=\sum_{n=0}^\infty a_n P_{2n+1}(x)$ , which is defined for $x$ in $[0,1]$, not for $-1$ to $1$.
I know the standard Fourier- Legendre ...
0
votes
1answer
27 views
summation of Legendre polynomials over a power law
I am trying to find a closed-form for the summation:
$$\sum_{n=0}^{\infty}\frac{P_{n}(x)}{(n+k)^{\alpha}}$$
where $P_{n}(x)$ denote the Legendre polynomials, k is a constant, and $\alpha$ is positive ...
1
vote
1answer
40 views
Series involving product of Legendre polynomials
I need to compute the following sum:
$$\sum_{n=0}^{\infty} (4n+3) P_{2n+1}(x)P_{2n+1}(y)$$
where $P_n(x)$ are the Legendre polynomials. Can anyone help me?
1
vote
0answers
21 views
Determining whether a singular endpoint is of limit-circle or limit-point case.
The ODE is $−[(1−x^2)u′]′-\mu u=f(x)$ over the interval $[-1,1]$.
I want to determine whether the singular endpoint $x=1$ is of a limit circle or limit point type. To do so, we were given a Theorem ...
1
vote
0answers
32 views
Finding the Legendre Polynomial formula from the Legendre equation
First I took the Legendre equation:
$$(1-x^2)\frac{d^2P_n(x)}{dx^2}-2x\frac{dP_n(x)}{dx}+n(n+1)P_n(x)=0$$
Then I wrote: $$P_n(x)=\sum_{k=0}^{n}a_{n, k} x^k$$
Where $a_{n, k}$ just gives the ...
3
votes
1answer
123 views
How to prove generating function of legendre polynomials? What am I doing wrong?
I am trying to prove the generating function of Legendre Polynomials:
$$g(x,t) = \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n$$
By only using the Legendre Differential Equation:
$$
(1-x^2)\...
0
votes
0answers
17 views
Legendre polynomial expansion of ellipsoid radius?
I am working on the following problem: I want to expand the radius of a generic ellipsoid in terms of Legendre polynomials. Say I have an ellipsoid:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^...
0
votes
1answer
20 views
Integral of product of Legendre polynomial $P_n(x)$ and shifted Legendre polynomial $\widetilde{P}_m(x)$
Does the integral
$$C_{n,m}\equiv\int_0^1P_n(x)\widetilde{P}_m(x)dx$$
have any closed answer? $P_n(x)$ is the Legendre polynomial of order $n$. $\widetilde{P}_m(x)\equiv P_m(2x-1)$ is called the ...
0
votes
1answer
31 views
Legendre polynomials and inner product
I have this Legendre polynomial:
$p_k(x) = \frac{1}{2^k k!} \frac{d^k}{dx^k} ((x^2 - 1)^k) $
I have calculated that: $p_0 = 1, p_1 = x, p_2 = \frac{1}{3}(3x^2-1), p_3 = \frac{1}{2}(5x^3-3x) $
Now I ...
2
votes
0answers
18 views
Integrate Jacobi polynomials $w_{a+\epsilon,b}(x)P^{a,b}_k(x)P^{a,b}_l(x)$ with slightly modified weight
It is well-known that the Jacobi polynomials satisfy the orthogonality relation
$$\int^1_{-1}w_{ab}(x)P^{a,b}_k(x)P^{a,b}_l(x)dx=\frac{2^{a+b+1}\Gamma(k+a+1)\Gamma(k+b+1)}{(2k+a+b+1)\Gamma(k+a+b+1)k!}\...
0
votes
0answers
34 views
How to derive general formula for $\int_0^1P_l(cos\theta)d(cos\theta)$
how do I derive general expression for this Legendre integrals on the path $(0,1)$ $$\int_0^1P_l(cos\theta)d(cos\theta)$$
P.S
Of course we can subtitude $cos\theta=x$ it doesn't really matter
0
votes
1answer
33 views
Legendre polynomial integration, basic
I want to evaluate $$\int_0^1x^2P_n(x)\,dx$$ but the only way out i see is by using $$xP_n(x)=\frac{nP_{n-1}+(n+1)P_{n+1}}{2n+1}$$ twice and since i got many $\int_0^1P_l(x)dx$, i use $$P_l(x)=\frac{...
1
vote
1answer
78 views
Legendre polynomial expansion
The known magnetic field of a simple loop $(B_z,B_r)$ is expressed as a combination of constant current I and the 2 Elliptic integrals K(k), E(k), k being related to the cylindrical position $(z,r)$ ...
4
votes
0answers
62 views
Series of Legendre Polynomials and Harmonic numbers. $\sum_{n=1}^{\infty} P_n (z) \frac{H(n)}{n+k}$
I would like to compute sums of the type
\begin{equation}
\sum_{n=1}^{\infty} P_n (z) \frac{H(n)}{n+k}
\end{equation}
where $P_n(z)$ are Legendre polynomials, $H(n)$ are harmonic numbers and $k = 0, 1,...
1
vote
0answers
70 views
Legendre polynomials satisfying a recurrence relation.
Monic Legendre polynomials (which are orthogonal polynomials) on $[-1,1]$ are defined as follows:
$$p_{0}(x)=1, \ p_{1}(x)=x$$
and $p_{n}(x)$ is a monic polynomial of degree $n$ such that $$\int_{-1}^{...
0
votes
0answers
27 views
Differentiation of a product involving divided differences
In 'Collocation at Gaussian Points' by DeBoer (top of p601):
We have a function $h_{t}(u) = G(t, u)r[\tau_{1}, ..., \tau_{k}, u]$ being considered on an interval $[a, b]$ where: $G(t, u)$ is a Green's ...
0
votes
0answers
22 views
Proving that Legendre polynomials decreasing about $n$ as $x$ approaches to 1.
I am in the study of the Legendre polynomials and think about proving
$$P_n(x)>P_{n+1}(x)$$ when $x$ is very close to 1.
This is obvious when I see this picture on Wikipedia. I was wonder if there ...
1
vote
1answer
23 views
Why is the Legendre polynomials eigenfunctions to this operator?
The operator $$Au = -\frac{d}{dx}((1-x^2)\frac{du}{dx}), \quad -1<x<1$$
supposedly has eigenfunctions
$$P_0 = 1, \quad P_1 = x, \quad P_2 = \frac{3}{2}x^2-\frac{1}{2}x$$
with eigenvalues
$$\...
0
votes
1answer
38 views
Legendre Polynomial Identity
I was looking for a method to do the following integral:
$\int^1_{-1}(1-x^2)\frac{dP_m(x)}{dx}P_n~dx$
I know there should be an explicit representation for the result but I am struggling to work it ...
0
votes
0answers
21 views
Formula for the derivative of finite power series in reversed order of terms.
I wanted to solve the polar part in Schrödinger's wave equation for the H-atom by direct substitution of functions of form:-
$$
\Theta_{lm}(\theta) = a_{lm} \sin^{|m|}\theta \sum_{r≥0}^{r≤(l-|m|)/2}(-...
0
votes
1answer
49 views
Alternative definition of Legendre polynomials
I'm studying Panofsky and Phillips' Classical Electricity and Magnetism. In writing the potential of a linear $2^n$-pole lying along the $x$-axis, they make use of the following definition for the ...
2
votes
3answers
74 views
How to simplify a partial sum obtained by Legendre polynomials
Context
I am working on an electrostatics problem. I have undertaken Fourier analysis. By $k$, I denote a natural number $k=0,1,2,\ldots$. I have obtained the following partial sum in terms of the ...
0
votes
1answer
52 views
Shifted Legendre polynomials symmetry relation
I have to prove that $p_n(1-x)=(-1)^np_n(x)$ ($x\in[0,1]$) for all $n\in\mathbb{N}$, where $(p_n)_n$ is the family of Legendre polynomials on $[0,1]$: given $(x^n)_{n=0,1,\ldots}$, $(p_n)_n$ is the ...
1
vote
0answers
42 views
Expand Associated Legendre Polynomials in the basis of shifted Associated Legendre Polynomials
I'm tasked to solve the following integral:
$$\int_0^{\pi/2}P^m_n(cos(\theta))P^m_l(cos(\theta))sin(\theta)d\theta$$
where $n,k\in Z^+$ and $m \in Z^{0+}$. Is it possible and wise to perform the ...
2
votes
1answer
56 views
Integral of Legendre polynomials with tangent
I have encountered the following relationship$^{[1][2]}$, stated without proof both times
$$\int_0^\gamma dt \tan(t/2)\cdot [P_n(\cos(t))+P_{n-1}(\cos(t))]=\frac{1}{n}[P_{n-1}(\cos(\gamma))-P_{n}(\cos(...
2
votes
1answer
21 views
Behavior of Legendre polynomials $P_\ell(\cos\theta)$ under $\theta\to\pi-\theta$
I'm studying some notes on the hydrogen atom which define the Legendre polynomials via Rodrigues's formula, $$P_\ell(z)=\frac{1}{2^\ell \ell!}\frac{d^\ell}{dz^\ell}(z^2-1)^\ell,\quad z=\cos\theta.$$ I'...
1
vote
0answers
91 views
Spherical Harmonics Sum Identity
I'm taking a course in Quantum Mechanics and this problem is causing me some struggles. Can someone help me prove this identity?
$$\sum_{m = -l}^l m^2 |Y_{l}^{m}(\theta, \phi)|^2 = \frac{l(l+1)(2l+1)}{...
1
vote
0answers
32 views
Plotting Legendre Functions
Quick question. In Griffith's Introduction to Quantum Mechanics, he introduces the associated legendre function $P_{l}^{|m|}(Cos \, \theta)$ and then proceeds to plot them on a polar plot. In doing so,...
5
votes
2answers
255 views
The coefficient and asymptotic in generating function
Let $L_n$ be the set of all the paths from $(0,0)$ to $(n,0)$ such that every step is $u=(1,1)$ , $d=(1,-1)$ and $r=(2,0)$.
Notice that the path could go under the $x$ axis.
a. Write a generating ...
1
vote
0answers
36 views
Integral form of legendre polynomial of second kind
I found a definite integral form of Legendre polynomial of second kind,
$$
Q_{n}(z)=\frac{1}{2}\int^{+1}_{-1}\frac{P_{n}(t)}{z-t}dt
$$
when n is an integer.
I wonder how to evaluate this integral. I ...
2
votes
1answer
129 views
Prove $1+\sum_{k=1}^{p} \frac{(-1)^k.n(n-1)(n-2)\cdots(n-2k+1)}{2^k.k!.(2n-1)(2n-3)\cdots(2n-2k+1)}=\frac{2^n(n!)^2}{(2n)!}$
Prove that $$
a_n\bigg[1-\frac{n(n-1)}{2(2n-1)}+\frac{n(n-1)(n-2)(n-3)}{2\cdot4\cdot(2n-1)(2n-3)}-\cdots+\frac{n(n-1)(n-2)\cdots(n-2k+1)}{2\cdot4\cdots 2k\cdot(2n-1)(2n-3)\cdots(2n-2k+1)}\bigg]=1\\
\...
2
votes
0answers
99 views
Question on the method of Gauss-Legendre
The Gauss-Legendre method seems to be one of the most important methods in solving differential equations numerically. I have a few questions related to it:
My teachers always assumed - without a ...
1
vote
0answers
82 views
Coefficient of the Highest Degree in the Power Series Solution to Legendre's Differential Equation
The Legendre's differential equation
$$(1-x^2)y''-2xy'+n(n+1)y=0$$
Substitute $y=\sum_{m=0}^\infty a_mx^m$
$$
\sum_{m=2}^\infty m(m-1)a_mx^{m-2}-\sum_{m=2} ^\infty m(m-1)a_mx^m-\sum_{m=1}^\infty ...
1
vote
0answers
23 views
Deriving normalization for the shifted associated Legendre function
Where can I find a solution for this integral:
$ \int_{a}^{b} P^m_l(c_1x + c_2b)P^{m'}_l(c_1x + c_2b)\,d(c_1x + c_2b)$, most solutions only solves for the interval [-1,1]. Of course I am looking for ...
1
vote
1answer
71 views
An asymptotic about Integral of Legendre Polynomials
I want to show asymptotics of the following integral involving Legendre Polynomial:
For $0<t<\theta<\frac\pi2$,
$$\Big|\int_0^\theta \frac{1}{\sqrt{\theta}+\sqrt{\theta-t}} \frac{1-P_n(\cos ...
2
votes
1answer
90 views
Integral of product of first derivatives of Legendre polynomials
I have the integral,
$$I=\int_{-1}^1 dx P'_m(x) P'_n(x).$$
I tried using $$P_m'(x)=-(m+1)\frac{xP_m(x)-P_{m+1}(x)}{x^2-1},$$ but that yields,
$$I=\int_{-1}^1\bigg(\frac{x^2P_mP_n}{(x^2-1)^2}+\frac{P_{...
2
votes
1answer
111 views
Legendre operator's eigenvalues and eigenfunctions
Consider the Lengendre operator which is a Sturm-Liouville operator defined in [-1,1] give as:
$$
\begin{equation}
\mathcal{L} u=-\frac{d}{d x}\left[\left(1-x^{2}\right) \frac{d u}{d x}\right]=-\left(...
0
votes
1answer
28 views
Legendre polynomials n=1 calculation by hand
Considering the Legendre polynomials:
$$P_n(x) = \sum_{m=0}^{n}a_{n,m}{x^m}$$
I know that: $P_0=1$ and $P_1=x$.
However given $P_0$ if I want to find $P_1$ by hand:
$$\langle P_1 | P_0 \rangle = 0 = \...
0
votes
0answers
20 views
Orthogonality legendre polynomials when $m = n$.
Now for the result when $n = m$, we use Rodrigues' Formula Theorem. Indeed,
\begin{equation}\label{eq:Pn2}
\int_{-1}^{1}(P_n)^2dx = \dfrac{1}{(n!)^22^{2n}}\underbrace {\int_{-1}^{1}\dfrac{d^n}{dx^n}(...
