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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Show that $K(u)=\sum_{k=0}^r P_k(0) P_k(u) \mathbf{1}_{\{|u| \leq 1\}}$ is a kernel of order $r$
I have a question concerning the construction of kernels wit orthogonal polynomials. The instructor defined the orthogonal polynomials as
$$P_0(x)=\frac{1}{\sqrt{2}}, P_m(x)=\sqrt{\frac{2 m+1}{2}} \...
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Legendre polynomials: computing $\int_0^1 xP_n(x)\,dx$
I am trying to find the integration:
$$
\int_{0}^{1}{xP_n\left(x\right)}\,dx
$$
I know that I should split Rodrigues's formula up into two parts $P_{2n}\left(x\right)$ for even terms and $P_{2n+1}\...
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further simplification of a summation involving Legendre and Associated Legendre polynomials
Was calculating something from a physics problem and found myself dealing with the following summation:
$$ \sum\limits_{\text{odd} \text{ } l}^{\infty}P_{l+1}(0)P'_l (\cos{\theta})\sin{\theta}\ $$
By ...
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Generating function of orthogonal polynomial basis
I'm studying the bases made up by orthogonal polynomial such as: Hermite, Legendre, Laguerre, Chebyshev. On my book there is a theoretical introduction that gives the difinition of generating function ...
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Integration of a square of Conical (Mehler) function
I want to evaluate
$$\int_{\cos\theta}^1 \left( P_{-1/2+i\tau}(x) \right)^2 dx,$$
where $P$ is the Legendre function of the first kind, $i$ is the imaginary unit, and $\tau$ is a real number.
Are ...
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Answer Check: Best Approximation of $(x+1)e^{-x}$ using Legendre Polynomials
I have worked through and produced an answer for the following question but am unsure of whether or not it is correct. I would appreciate some insight.
$\textbf{Question}$:
Using the Gram-Schmidt ...
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Proof of Legendre Polynomials In A Differential Equation
I'll keep this question as math-based as possible while avoiding any physics. In quantum mechanics, solving the time-independent Schrodinger equation in polar coordinates involves using separation of ...
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Averaged value of product of Legendre Polynomials
Note: the following question comes from Alex Meiburg via Faceboook and was found via his work with the Legendre Polynomials in quantum machine learning.
Let $P_k$ be the $k$-th Legendre Polynomial.
...
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Can order of summation and integral be interchanged in : $\int_{-1}^1 ( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime))\text{d}\xi^\prime$?
I am wanting to know if there is a proof that the order of summation and integration can be interchanged in $$\int_{-1}^1 \left( \sum_{n=0}^\infty P_n(\xi)P_n(\xi^\prime)Q_l(\xi^\prime)\right)\text{d}\...
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Legendre polynomial as basis for finite element method
When people say use Legendre polynomial as basis polynomial in fem, are they using the polynomials themselves or the integral of those polynomials?
I'm asking this because I have this poisson equation:...
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Confusion about spherical harmonics, Legendre polynomials
I'm quite new to the ideas behind spherical harmonics and Legendre polynomials. I have a couple of questions about them.
Spherical harmonics, as I understand them, are functions that can be used to ...
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How to integrate products of Legendre functions over the interval [0,1]
The associated Legendre polynomials are known to be orthogonal in the sense that
$$
\int_{-1}^{1}P_{k}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{k,l}
$$
This is intricately linked to ...
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Series representation of the associated Legendre polynomial
I have found the following identity for the associated Legendre polynomial to be true:
$$
P_{n}^{m}(\tau)=\frac{n!(n+m)!}{2^n}\sum_{s=0}^{n-m}\frac{(-1)^{m+s}(1+\tau)^{n-m/2-s}(1-\tau)^{m/2+s}}{(n-s)!...
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Any comment to speed up the calculation of double-integral having Legendre polynomials?
I want to compute the following double integral at $t=t_{0}$ rapidly. I tried different methods, but all are time consuming for I,J,M >7.
Any comments to speed up the calculation???
$$\frac{1}{2}\...
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Evaluating Legendre $\int_a^b P_{2n}(\text{Ei}(x))dx,\int_a^b P_{2n}(\text{li(x)})dx$ to solve li$(\mu)=0$ with logarithmic/exponential integral
$\def\li{\operatorname{li}}\def\Ei{\operatorname{Ei}}\def\P{\operatorname P}\def\W{\operatorname W}$
The Soldner Ramanujan constant $\mu$ is possibly expressible via a Dirac $\delta(x)$ Legendre $\P_v(...
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Getting the recurrence relation for Legendre polynomials by Leibnitz rule
Exercise:
For each natural number $n$ define
$$\phi_n(x)=\frac{d^n}{d x^n}\left(x^2-1\right)^n$$
Derive the formulas
$$\phi_{n+1}^{\prime}(x)=2(n+1) x \phi_n^{\prime}(x)+2(n+1)^2 \phi_n(x)$$
$$\phi_{n+...
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Inner product of 4 Legendre Polynomials
Is there a closed form for the quadruple inner product of Legendre Polynomials such as:
\begin{align}
\int_{-1}^{1}P_k(x)P_l(x)P_m(x)P_n(x)dx
\end{align}
I am aware of solutions for the triple inner ...
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Intuitively understanding meaning of "normalization constant"
I am reading real analysis book. It is written in the text that the functions $f_n$ are (up to a normalisation constant) the Hermite polynomials. What does it mean "up to a normalisation constant&...
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How to calculate Integral of sine raised to the power of $2l+1$ [duplicate]
I encountered the following integral when dealing with Legendre polynomials, trying to derive their orthonormality relation starting from the Rodriguez Formula.
$$\int_0^{\pi/2}\sin(\theta)^{2l+1}d\...
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How to merge odd series and even series of hypergeometric function of Legendre polynomials into one hypergeometric function?
On the Wolfram MathWorld page of Legendre Differential Equation, Legendre polynomials are represented as
$$
P_l(x) = c_n
\begin{cases}\begin{align*}
&_2F_1\left(-\frac{1}{2}(l), \frac{1}{2}(l + 1);...
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The $n$th derivative of the $n$th spherical Bessel function
I quote Problem 12.4.7 of the 5th edition of Mathematical Methods for Physicists by Arfken, Weber, and Harris:
A plane wave may be expanded in a series of spherical waves by the Rayleigh equation:
$$
...
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How to derive the three-term recurrence relation for the orthogonal polynomials associated with the Legendre-Gauss-Lobatto quadrature?
I wrote a computer code to calculate the inner nodes (i.e. excluding the end points) in the Legendre-Gauss-Lobatto (LGL) quadrature rule ~5 years ago by solving for the eigenvalue of a certain ...
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Proof of Bonnet's Recursion Formula for Legendre Functions of the Second Kind?
I'm doing some self-study on Legendre's Equation. I have seen and understand the proof of Bonnet's Recursion Formula for the Legendre Polynomials, $P_n(x)$.
$$(n+1)P_{n+1}(x) = (1+2n)xP_n(x) - nP_{n-...
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Prove that polynomial $p(x)$ od degree $n$ is equal to $cP_{n}(x)$, where $P_{n}(x)$ is Legendre polynomial.
This is the problem $5$ from chapter $45$ on properties of Legendre polynomials from Simmons book "Differential Equations with Applications and Historical Notes".
If $p(x)$ is a polynomial ...
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Integral connection with Hermite and Legendre polynomials
Show that $$\int\limits_{-\infty}^{+\infty}x^n e^{-x^2} H_n(tx) dx =\sqrt{\pi} n! P_n(t)$$
Case seems rather complex, I'm completely stuck...
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Could someone help prove some properties of Legendre Polynomials?
I have already proved other properties of the Legendre polynomials, like:
$$P_n(-x) = (-1)^n \, P_n(x)$$
$$P_{2n+1}(0) = 0$$
$$P_n(\pm1)= (\pm1)^n$$
But I can't get this one:
$$P_{2n}(0) = \frac{(-1)^...
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Distributional Fourier Transform of $P_{n}(\cos x)$
Consider the following set of Fourier transforms (understood as distributions since formally these are divergent):
$$
D_{n}(y) := \frac{1}{2\pi} \int_{-\infty}^{\infty} P_{n}(\cos x) e^{i x y} dx
$$
...
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A closed-form formula involving Legendre polynomials for the "information potential" of the binomial distribution
This is a follow-up of this recent question to which I have given an answer.
It is mainly the "Edit" part of this answer that has an interest here with the following formula I have found (...
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How to prove Legendre Polynomials' recurrence relation without using explicit formula?
Assume there is an inner product on linear space $V = \{ \text{polynormials}\}$:
$$\langle f, g\rangle = \int_{-1}^1 w(t) f(t)g(t) dt$$ with $w(t) \ge 0$ and not identically zero.
Then we can ...
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Prove of a Legendre's polynomial problem
I have a question that I need to prove that for $n≥1$,
$$\frac{1}{2n}\int_{-1}^{1}x\frac{d}{dx}(P_n(x)^2)dx=\frac{2}{2n+1}$$
I have to evaluate the integral instead of using the orthogonality property ...
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Prove $\int_{0}^{1}x^{m}P_{l}(x)dx = \frac{m! (m - l + 1)!!}{(m - l +1)!(m + l +1)!!}$
I used the Rodrigues formula and integrated by parts:
$$ \dfrac{1}{2^{l}l!}\int_{0}^{1}x^{m}\dfrac{d^{l}}{dx^{l}}(x^2 - 1)^{l}dx = \dfrac{1}{2^{l}l!}[ x^{m}\dfrac{d^{l-1}}{dx^{l-1}} (x^{2} - 1)^{l} - \...
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Prove: $\int_{-1}^1 x^n P_n(x) dx = \frac{2^{n+1}n!^2}{(2n+1)!}$
I'm trying to prove:
$$\int_{-1}^1 x^n P_n(x) dx = \frac{2^{n+1}n!^2}{(2n+1)!}$$
My attempt consisted in applying the Rodrigues formula:
$$\int_{-1}^1 x^n P_n(x)dx = \dfrac{1}{2^n n!} \int_{-1}^1 x^n \...
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Combinatorial interpretation of the coefficients of Legendre polynomials?
For example, in the Hermite polynomial $\operatorname{He}_n(x)$, the absolute value of the coefficient of $x^k$ is the number of (unordered) partitions of an $n$-element set into $k$ singletons and $\...
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Prove that $\sum_{y=0}^{p-1}\left(\frac{y}{p}\right)\left(\frac{y+d}{p}\right)=-1$
The question is from The number of solutions of $ax^2+by^2\equiv 1\pmod{p}$ is $ p-\left(\frac{-ab}{p}\right)$
And I wonder how to prove the following equation although someone gives the trick:
\begin{...
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Prove that $\int_{0}^{1} P_{l}(x)dx = \dfrac{P_{l-1} (0)}{l + 1}$
I've tried to prove this through the recurrence relation
$$ lP_{l}(x) + P'_{l-1}(x) - xP'_{l}(x) = 0 \rightarrow l\int_{0}^{1} P_{l}(x) dx + \int_{0}^{1} P'_{l-1}(x) dx - x \int_{0}^{1} P'_{l}(x)dx = ...
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Convergence of an integral with Legendre polynomials
Let's consider the following integral
$$
I(\ell) = \int_{-1}^1 dx P_\ell(x) A(x)
$$
where $ P_\ell(x) $ is the $\ell$-th Legendre polynomial and $A(x) = \frac{1}{1-\lambda x}$ with $0\leq \lambda<1$...
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Deriving the integral form of the Legendre polynomials from the derivative form
I have the Legendre polynomials $P_n(x)= \frac{1}{n!2^n}\frac{d^n}{dx^n} (x^2−1)^n$ and have to show/derive the integral form $P_n(x)=\frac{1}{\pi} \int_0^{\pi}(x+\sqrt{1−x^2}\cos t)^n dt$ for $|x|<...
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Simplification of an infinite sum consisting of Legendre polynomials
In an article about Legendre Polynomials, I encountered the following simplification.
\begin{align}
(something)\dots&=\int_{-1}^{1} \int_{-1}^{1} \left[\sum_{i=n+1}^{\infty} \sum_{j=n+1}^{\infty} ...
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Integration of product of two Legendre polynomial
I am trying to understand the Integration of the product of two Legendre polynomials.
Given,
$$I=\int_{-1}^{1} P_{l}(x)P_{l'}(x)\mathrm{d}x~$$
Suppose $x = \cos(\theta)$, then our $I$ becomes
$$I=\...
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Reference(book or article) for an explicit formula of Legendre polynomials
The following explicit formula is stated for Legendre polynomials on Wikipedia.
\begin{equation}
P_n(x)=\sum_{k=0}^n {n\choose k}{n+k \choose k} \left(\dfrac{x-1}{2}\right)^2
\end{equation}
Do you ...
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How do we determine if an operator over real functions is normal?
We have the operator $T(f) = (pf')'$, where $p(x) = x^2 - 1$. The inner product is $\displaystyle (f,g) = \int_{-1}^1 f(x)g(x) dx$. How do we infer whether eigenfunctions corresponding to different ...
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Uniqueness of the nodes for Gauss-Legendre quadrature
Gauss-Legendre quadrature approximates $\int_{~1}^{1}f(x)dx$ by $\sum_{i=1}^nw_if(x_i)$.
Wikipedia says that
This choice of quadrature weights $w_i$ and quadrature nodes $x_i$ is the unique choice ...
user1095295
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Relationship between Gauss–Legendre quadrature and continued fractions
Wikipedia says the following:
Carl Friedrich Gauss was the first to derive the Gauss–Legendre quadrature rule, doing so by a calculation with continued fractions in 1814.
https://en.wikipedia.org/...
user1095295
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What will be the explicit formula for Shifted Legendre's polynomial in interval $x\in[a,b]$
If I defined shifted Legendre polynomial $\tilde P_{n}(x)=P_{n}(\frac{2x-b-a}{b-a}) for\;all x\in[a,b]$ Then what will be the explicit formula $P_{n}(x)=\sum_{k=0}^{n} (-1)^{n+k} \frac{(n+k)!}{(n-k)! ...
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Expansion of divergent function with Legendre polynomials
Polynomials on $x\in[-1,1]$ can be written as an expansion in the Legendre polynomials $P_l(x)$. Is it possible to expand more general types of function on this interval in terms of these polynomials. ...
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where these formulas for Legendre polynomials and series come from?
I am reading a textbook on differential equations.The chapter I am studying now is about solving differential equations using series.The book solves Legendre's differential equation$$(1-x^{2})y''-2xy'+...
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Problem about the solution of Legendre's equation
I am reading a textbook on differential equations. This book is not written in English and surely you have not heard of it. The chapter I am studying now is about solving differential equations using ...
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Integrals of products of Legendre polynomials
Define
$$
\tilde{P}_{n}^{m}(\tau)=(1-\tau^2)\frac{d}{d\tau}P_{n}^{m}(\tau)
$$
where $m$ and $n$ are integers such that $|m|\leq n$ and $P_{n}^{m}(\tau)$ are the associated Legendre polynomials. How do ...
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1
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Find the best approximation with Legendre polynomials
Can you help me with this problem?
I have the Legendre polynomials $ψ_{k}(x):= \frac{k!}{(2k)!}\frac{d^k}{(dx^k)}(x^2-1)^k$.
They are orthogonal with respect to the $L^2$ scalar product on $[−1, 1]$.
...
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Proof that $m \neq 0$ spherical harmonics terms vanish when expanding $f(\theta, \phi)$, $\theta = 0$
I am a physics undergraduate. I'm working through some sections of 3ed Jackson Electrodynamics. I am on a section discussing what are called spherical harmonics:
$$Y_{ml}(\theta, \phi) = CP^m_l(cos\...
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