Skip to main content

Questions tagged [legendre-polynomials]

For questions about Legendre polynomials, which are solutions to a particular differential equation that frequently arises in physics.

Filter by
Sorted by
Tagged with
1 vote
1 answer
51 views

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} ...
Sanjana's user avatar
  • 263
9 votes
0 answers
227 views

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, ...
Setness Ramesory's user avatar
1 vote
1 answer
40 views

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 0 1 2 3 4 0 0 2 0 2 0 1 0 ...
luyipao's user avatar
  • 47
2 votes
0 answers
43 views

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 ...
Xinhan Yuan's user avatar
1 vote
0 answers
23 views

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 ...
user12588's user avatar
  • 359
5 votes
2 answers
102 views

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 $...
bkocsis's user avatar
  • 1,258
1 vote
0 answers
42 views

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 ...
Purnendu's user avatar
1 vote
2 answers
99 views

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{...
Lee Davis-Thalbourne's user avatar
0 votes
1 answer
60 views

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.
azozer's user avatar
  • 17
2 votes
0 answers
33 views

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 ...
Oscar's user avatar
  • 901
0 votes
1 answer
50 views

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|} \; ...
Edward Henry Brenner's user avatar
0 votes
1 answer
43 views

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 ...
azozer's user avatar
  • 17
1 vote
0 answers
19 views

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_{-...
Suvajit Dey's user avatar
0 votes
0 answers
23 views

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 ...
Silver Pages's user avatar
3 votes
0 answers
63 views

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 ...
Álvaro Zorrilla Carriquí's user avatar
0 votes
0 answers
69 views

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 ...
HtmlProg's user avatar
0 votes
0 answers
40 views

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 ...
Beast's user avatar
  • 11
5 votes
1 answer
207 views

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 ...
Math Attack's user avatar
0 votes
0 answers
37 views

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, \...
pmu2022's user avatar
  • 194
0 votes
0 answers
46 views

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 ...
markusas's user avatar
  • 346
2 votes
0 answers
62 views

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} ...
Clara's user avatar
  • 29
0 votes
0 answers
35 views

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= \...
giladude's user avatar
  • 973
0 votes
1 answer
35 views

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.
Wyel Spinor's user avatar
2 votes
2 answers
90 views

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 $n$, ...
Jog's user avatar
  • 357
0 votes
0 answers
22 views

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?
Ajay Mehra's user avatar
1 vote
1 answer
103 views

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 = \...
Habouz's user avatar
  • 366
1 vote
0 answers
17 views

Integrals of Legendre polynomial and a rational function

Is there are analytic expression of the following definite integral? $$ \int_{-1}^1 x^\alpha (1-x^2)^{\beta} P_l(x) P_m(x) \text{d}x $$
Lyle Kenneth Geraldez's user avatar
2 votes
0 answers
21 views

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 (...
Hakan Akgün's user avatar
0 votes
0 answers
42 views

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 ...
Rosstopher's user avatar
0 votes
0 answers
26 views

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_\...
Lyle Kenneth Geraldez's user avatar
1 vote
1 answer
61 views

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 \...
javery's user avatar
  • 55
2 votes
1 answer
78 views

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 ...
Angus0517's user avatar
0 votes
1 answer
80 views

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 ...
KenN's user avatar
  • 21
0 votes
0 answers
19 views

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 ...
Marcosko's user avatar
  • 173
0 votes
1 answer
100 views

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 ...
Anon's user avatar
  • 11
1 vote
0 answers
21 views

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'}...
MkFlash's user avatar
  • 19
1 vote
1 answer
108 views

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 $...
Addem's user avatar
  • 5,706
0 votes
0 answers
22 views

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(...
user1151712's user avatar
1 vote
1 answer
62 views

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 ...
rlarson's user avatar
  • 13
0 votes
0 answers
73 views

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 $...
knuth's user avatar
  • 31
0 votes
0 answers
78 views

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}{...
user avatar
0 votes
0 answers
33 views

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 ...
ExcitedSnail's user avatar
2 votes
1 answer
126 views

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\...
Max's user avatar
  • 546
0 votes
0 answers
75 views

"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 ...
Luke's user avatar
  • 203
2 votes
0 answers
221 views

Integral of associated Legendre polynomials over the unit interval

I am looking for a closed-form expression for the integral of the associated Legendre polynomial $P_l^m$ over the unit interval ($l \ge m$ non-negative integers), $$ I_l^m = \int_{0}^{1} P_l^m(x) \, ...
ntessore's user avatar
1 vote
1 answer
553 views

On the vanishing of the integral $\int^{n+4}_0 dt P_{\ell}\left(1 - \frac{2t}{n+4}\right)(-t+2)_{n+1}$ for $\ell \geq n+2$

I came across the following integral \begin{equation} \int^{n+4}_0 dt P_{\ell}\left(1 - \frac{2t}{n+4}\right)(-t+2)_{n+1} \, , \end{equation} where in the above $P_{\ell}(x)$ is the Legendre ...
user avatar
2 votes
1 answer
2k views

First derivative of Legendre Polynomial

Legendre polynomials ($P_n$) are defined as a particular solution to the ODE. $$(1-x^2)P_n^{''}-2xP_n^{'}+n(n+1)P_n=0$$ It is expressed by Rodrigues’ formula. $$P_n=\frac{1}{2^nn!}\frac{d^n}{dx^n}((x^...
Il Prete Rosso's user avatar
0 votes
0 answers
29 views

Formula for Transformation of Polynomial Coefficients under Rotation

I have a function represented in a basis of 2D Legendre polynomials, $$ f(x,y) = \sum_{n=0}^N c_n P_n(x,y) $$ where $P_n(x,y)$ is a 2D Legendre polynomial given by $$ P_n(x,y) = P_l(x)P_m(y) $$ where $...
David G.'s user avatar
  • 260
2 votes
1 answer
282 views

How to obtain the Legendre Polynomials from the power series solution (of the Legendre's equation)?

Solve the differential equation $$(1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+n(n+1)y=0.$$ Show that a polynomial, say $P_n(x)$ is a solution of the above equation, when $n$ is an integer. I tried ...
Thomas Finley's user avatar
-1 votes
1 answer
59 views

Legendre's Series: How can I show that $\int_{-1}^1 f(x)S(x)dx= f(1)$

Consider the series: $$S(x)=\sum_{n=0}^{\infty}\frac{(2n+1)P_n(x)}{2}$$ Show that: $$\int_{-1}^1 f(x)S(x)dx= f(1)$$ where $f(x)$ is any function of the interval $[−1, 1]$ on the real numbers which can ...
tom.2023's user avatar

1
2 3 4 5
13