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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Legendre addition theorem in $2$ dimensions
We know the addition theorem for Legendre polynomials in spherical coordinates is
$$P_\ell(\cos\gamma)=\dfrac{4\pi}{2\ell +1}\sum_{m=-\ell}^\ell\mathrm Y_{\ell m}(\theta_1,\phi_1)\,\mathrm Y_{\ell m}^\...
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Proof involving integrals, binomial coefficients and Legendre polynomials
I'm currently working on a research paper involving L-moments statistics. This is the first time for me working on that subject.
I stumbled upon this article mentioning a very important equality, ...
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Deriving quadrature weights from discrete orthogonality of exponentials
In the proof of Lemma 2 of Driscoll and Healy, it says
\begin{align}
\sqrt{2}\delta_{k,0} &= \frac{1}{\sqrt{2}}\int_0^\pi P_k(\cos\theta)\sin\theta d\theta\\\\
&= \frac{1}{2\sqrt{2}}\int_{-\pi}...
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Derivation of Legendre Polynomials from only orthogonality
I recently stumbled on the idea of Legendre polynomials, from the perspective of using them to approximate functions over a region, and I discovered all of these other ways to express them, using ...
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Derivation of the associated Legendre Polynomials
I have been struggling to find a proper derivation of the associated Legendre Polynomials and a derivation of
$$P_l^{-m}(\mu)=(-1)^m\frac{(l-m)!}{(l+m)!}P_l^m(\mu)$$
Can someone point to a proper ...
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Could someone explain the reason behind using Legendre Polynomials?
Where $x = \cos(\phi)$ and therefore (and this is very important) $x=[-1,1]$, we have the (special such that m = 0) associated Legendre Equation $(1-x^2)\frac{d^2 y}{dx^2} - 2x \frac{dy}{dx}+ky = 0$, ...
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How to find an expression for the $n$th partial derivatives of $1/r$?
From Pirani's lectures on General Relativity I got the following identity which he asks the reader to prove by induction which is easy
$$\partial_{\alpha_1} \partial_{\alpha_2} ... \partial_{\alpha_n} ...
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Evaluate $\int_{0}^{1} \operatorname{Li}_3\left [ \left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ] \text{d}x$
Possibly evaluate the integral?
$$
\int_{0}^{1} \operatorname{Li}_3\left [
\left ( \frac{x(1-x)}{1+x} \right ) ^2 \right ]
\text{d}x.
$$
I came across this when playing with Legendre polynomials, ...
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Integration of Legendre polynomials with their derivatives
I need the results of $\int_{-1}^1 p_i(x) p_j'(x) dx$, where $p_i$ is the classical Legendre polynomials in $[-1,1]$. Using the matlab, I get the following result:
i \ j
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Prove the orthogonality of the Legendre Polynomial from the recursion only.
It's known that the Legendre Polynomials follow the recursion:
$$P_n(x)=\frac{2n-1}{n}xP_{n-1}(x)-\frac{n-1}{n}P_{n-2}(x)$$
with
$$P_0(x) = 1, P_1(x)=x$$
Now I am finding an elementary method to prove ...
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Tripe integral involving the square of associated Legendre polynomials and a derivative of a Legendre polynomial
I encountered the following integral in the physics literature
$$
\int_{-1}^{1}P_{\ell}^m(x)^2P_{n}^\prime(x){\rm d}x
$$
where $P_{\ell}^m(x)$ is an associated Legendre polynomial of degree $\ell$ and ...
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Simplify in closed-form $\sum_n P_n(0)^2 r^n P_n(\cos \theta)$
Simplify in a closed form the sum $$S(r,\theta)=\sum_{n=0}^{\infty} P_n(0)^2 r^n P_n(\cos \theta)$$ where $P_n(x)$ is the Legendre polynomial and $0<r<1$. Note that $P_n(0)= 0$ for odd $n$ and $...
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Integral of product of Legendre polynomial and exponential function
Kindly help me with the following integral :
$
I_l(a) = \int_{-1}^{+1} dx\, e^{a x} P_l(x) \quad
$
($a$ is real and positive).
I thought to use the following relation given in Gradshteiyn and also ...
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Calculation for negative integer order Associated Legendre Function
I am currently engaging with the following hypergeometric function as a result of attempting to find a solution for this probability problem for $n$ number of dice:
$$_2F_1\left (\frac{n+k}{2}, \frac{...
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Is the Gram-Schmidt Orthogonalization process for functions the same as it is for vectors? [closed]
I was not able to find resources online and was wondering so since it would greatly help me with my work if I can directly apply what I have learned from linear algebra.
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Multidimensional Legendre polynomials?
Legendre polynomials can be given by several expressions, but perhaps the most compact way to represent them is by Rodrigues' formula as
$$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n.$$
I ...
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Morse and Fesbach identity $\sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})|u|} \; P_n(x) =(2\cosh u - 2x)^{-1/2}$
In book called Methods of theoretical physics from Morse and Feshbach, there is identity, which I wanted to prove ($P_n(x)$ are Legendre polynomials):
$$\sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})|u|} \; ...
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Finding $l$ such that the Legendre differential equation has a polynomial solution
I was given this problem for practice and was wondering if my approach was correct:
$$
(1-x^2)y'' - 2xy' + 3ly = 0.
$$
At first I thought of just using $l = 2$ since the Legendre DE is defined in ...
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Is it possible to prove orthogonal form of integral of legendre polynomial solely from legendre's differential equation without using anything?
The differential equation for the Legendre polynomials
$P_n(x)$ is given by:
$(1 - x^2) \frac{d^2P_n}{dx^2} - 2x \frac{dP_n}{dx} + n(n + 1)P_n = 0$. Now I want to prove that $\begin{equation} \int_{-...
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Investigating the numerical accuracy of a truncated Legendre polynomial expansion of an unknown function
I have an integral equation involving an unknown function $f(x)$, of the most basic form
$$
\int_{-1}^{1} e^{iω(t) x} f(x) \ dx = g(t)
$$
I am solving for an approximation of $f(x)$ by substituting in ...
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Calculating the behaviour of an integral with Legendre polynomials of large order [closed]
I need to calculate the following integral:
$$\int_{\theta, \phi \in S^2} \left [ P_\ell(1-2\sin ^2\theta \sin^2\phi) \right ]^2 \sin\theta\, d\theta\, d\phi$$
where $S^2$ represents the unit sphere ...
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Closed Forms for Sums of Legendre Polynomials
I am investigating a series of the form $$\sum_{n=0}^\infty \frac{1}{1 + e^{nx}}P_n(x)$$ where $P_n$ is the Legendre Polynomial of degree $n$.
Because I am dealing with many of these series, it would ...
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legendre solution for non homogenous equation
given the legendre equation $(1-x^2)y'' - 2xy' + by = f(x)$
why can the solution be a series of legendre polynomials $y(x) = \sum_{n=0}^{\infty}a_n P_n(x)$? i thought legndre solves the homogenous ...
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How to express a Gaussian as a series of exponential? $\displaystyle e^{-x^2}=\sum_{n=1}^{\infty}c_n e^{-nx}$
Context
I would like to express the Gaussian function as a series of exponentials:
$$e^{-x^2}=\sum_{n=1}^{\infty}c_ne^{-n|x|}\qquad\forall x\in\mathbb{R}$$
For simplicity (the absolute value is added ...
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Expansion of $\frac{\text{erfc}({|\vec{r} - \vec{r'}|})}{|\vec{r} - \vec{r'}|}$ in spherical harmonics?
How can I derive the spherical harmonic expansion coefficients for the function $$ \frac{\text{erfc}({|\mathbf{r} - \mathbf{r'}|})}{{|\mathbf{r} - \mathbf{r'}|}} $$ by expressing it as $$f(\theta, \...
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Interpolation and general Gaussian quadrature
I just finished a course on numerical mathematics, and have become quite interested in interpolation and how it ties into numerical integration. What suprised me while studiyng quadrature was the fact ...
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Fourier-Legendre series for $x^n$
I'm struggling to find the Legendre expansion for $x^n$ (exercise 15.1.17 from Mathematical Methods for Physicists).
I'm trying to evaluate the following integral:
$$a_m = \frac{2m+1}{2} \int_{-1}^{1} ...
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Product of d-dimensional Legendre polynomials
Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...
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How to caculate this integral by Legendre Poly.
How to caculate the integral
$$\int_{-1}^1(1-x^2)\mathrm{P}_k'(x)\mathrm{P}_l'(x)~\mathrm{d}x$$
Where $\mathrm{P}_l(x)$ is the $l$ - oeder Legendre Poly.
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Integral involving even order Legendre polynomials
Let $a>1$. We want to evaluate the integral
\begin{equation*}
\int_{-1}^1 \frac{P_{2n}(\xi)\,d\xi}{\sqrt{a^2-\xi^2}}
\end{equation*}
Mathematica is able to evaluate special cases for various $n$, ...
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Legendre Differential equation, n(n-1) or n(n+1)
I am confused regarding the Legendre Differential Equations' coefficients.
In some books its,
$(1-x^2)y''-2xy'+n(n-1)y=0$
and somewhere it is,
$(1-x^2)y''-2xy'+n(n+1)y=0$
what is its correct form?
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Legendre's Polynomial and spherical harmonics
The differential equation that is satisfied by the Legendre's polynomials is:
$$(1-x^2)y'' - 2xy' + \lambda y = 0 (*)$$
I have also been told that the Legendre's polynomial with the parameter $x = \...
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Integrals of Legendre polynomial and a rational function
Is there are analytic expression of the following definite integral?
$$
\int_{-1}^1 x^\alpha (1-x^2)^{\beta} P_l(x) P_m(x) \text{d}x
$$
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Understanding the Dual Emergence of Legendre Polynomials in Differential Equations and Orthogonalization
I am currently examining the mathematical properties of Legendre polynomials and have observed their emergence in two distinct areas: as solutions to a specific class of differential equations (...
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Legendre Polynomial Triple product with different arguments
I'm trying to integrate this: $$f_{jkl}\langle{\hat{a},\hat{b},\hat{c}}\rangle=\frac{1}{4\pi} \int d{\Omega}_n P_j(\hat{a}.\hat{n})P_k(\hat{b}.\hat{n})P_l(\hat{c}.\hat{n})$$ where $\hat{n}$ is a unit ...
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I want to prove the proposition that the absolute value of integral expression must be monotonically decreasing
For arbitrary $r_0$ and $P_l(\text{cos}(\theta))$ be the Legendre polynomials,
$$
E_n=\int_{r0}^\infty \int_0^\pi -\text{sin}^3(\theta)(\left(
\left(\text{cot}(\theta) \sum_{l=0}^n R_l(r) \partial_\...
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Generating Function for Bivariate Legendre Polynomials?
I am aware of the following standard generating function for single-variable Legendre Polynomials:
$$
\sum\limits_{n=0}^{\infty}P_n(x)z^n = \frac{1}{\sqrt{1-2xz+z^2}}
$$
for $x \in \mathbb{R}, z \in \...
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What is the combinatorics meaning of the generating function for Legendre polynomials?
I know the generating function has been a super useful tool when finding the Legendre polynomials (or other special functions), or even used to estimate the static electric potential. In the Physics ...
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Norm of Legendre Polynomials $P_m(x)$
While studying to prove the norm of Legendre polynomials $P_m(x)$ is $\sqrt{\frac{2}{2m+1}}$, I faced $\int_{-1}^{1} [D^m (x^2-1)^m]^2 dx = (2m)! \int_{-1}^{1} (1-x^2)^m dx.$ $D^m$ stands for ...
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Spheroidal eigenvalues with shifted boundary conditions
I was studying the spheroidal differential equation in relation to calculating solutions for fields in a general Kerr background metric and, as far as I can tell, the eigenvalues $\lambda$ that enter ...
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To expand $f(x)=\begin{cases}-1, &-1<x<0\\+1,&0<x<1\end{cases}$ in Legendre polynomial series and obtain formula for expansion coefficients
Expand the step function
$$f(x)=\begin{cases}-1, &-1<x<0\\+1,&0<x<1\end{cases}$$
in a series of Legendre polynomials $P_l(x)$. Obtain an explicit formula for the expansion ...
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Integral of Squared Spherical Harmonics
The following integral comes out of an expression $\langle |Y_{l,m}(\theta, \phi)|^2\rangle$ over a orientation probability distribution:
$$\int_{0}^{2\pi} \int_{0}^{\pi} Y_{lm}^2(\theta, \phi)Y_{l'm'}...
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Prove from the generating function that even index Legendre polynomials are even functions.
At this link: http://www.phy.ohio.edu/~phillips/Mathmethods/Notes/Chapter8.pdf
The author writes that one can prove from the generating function of Legendre polynomials that $P_{2n}$ are all even and $...
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Integral Formula Involving Legendre Polynomials
The following exercise takes the form;
$\int_{0}^{\infty}f\left(\frac{P_{n+1}\left(x\right)}{P_{n}\left(x\right)}\right)\cdot\frac{1}{P_{n}\left(x\right)^{2}}dx=\left(n+1\right)\int_{0}^{\infty}f\left(...
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How to prove this summation equation? [duplicate]
I'm looking for some hints on proving the following (either directly or by induction):
$$
\sum_{k={0}}^{l/2} \frac{(-1)^k(2l-2k)!}{k!(l-k)!(l-2k)!} =2^l
$$
I do know it is actually true from various ...
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Coefficient of $x^n$ in Legendre series expansion
Suppose we are approximating a function $f$ with a Legendre series of order $N$, namely
$$
f(x) \approx \sum_{n=0}^N c_n P_n(x) \equiv f_N(x)
$$
where $P_n(x)$ is the $n^{th}$ Legendre polynomial and
$...
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Re-writing a sum of binomial coefficients as an integral of shifted Legendre polynomials
This is a question regarding the answer presented here.
In order to make this post self-contained, I am wondering if someone can explain why the sum
$$
\sum_{k=0}^{n}(-1)^{n+k}\binom{n}{k}\binom{n+k}{...
user1211323
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How to construct Legendre polynomials for $x_1,...,x_k$?
I'm trying to run a nonparametric regression to estimate the unknown conditional mean $E(Y|X_1=x^*_1,X_2=x^*_2)$ using data set $\{Y_i,X_{1i},X_{2i}\}_{i=1}^n$. This could be done by nonparametric ...
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Asymptotic equality of $\frac{1}{n!}\frac{d^n}{dx^n}(x^n(1-x)^n) $
Consider the Shifted Legendre Polynomial $$\tilde P_n(x)=\frac{1}{n!}\frac{d^n}{dx^n}(x^n(1-x)^n) $$ where $n\in\mathbb{N}\cup\{0\}$
Question: What is the asymptotic equality of $\tilde P_n(x)$ as $n\...
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"Legendre-type" integrals involving $\frac{dt}{\sqrt{t^2-2t\cos(\theta)+1}}$
Summing Legendre polynomials $P_{l}(\cos\theta)$ often leads to expressions containing $\frac{1}{\sqrt{t^2-2t\cos\theta+1}}$, as this is the generating function for the Legendre polynomials. I want to ...
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