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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4
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1answer
66 views

Fourier Legendre expansion of $_3F_2\left( 1,1,1;\frac32,\frac32;x\right)$

Problem: Can we obtain a closed-form Fourier-Legendre expansion of the following hypergeometric series? $$_3F_2\left( 1,1,1;\frac32,\frac32;x\right)$$ Or equivalently, how to evaluate $$I(n):=\int_0^1\...
0
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1answer
23 views

How to express in Legendre's polynomials?

How do I express $cos(3\theta)$ and $sin^{2}(\theta)$ in Legendre's polynomials, knowing that $x=cos\theta$? I know that $f(x)=\sum a_{n}P_{n}(x)$ and $P_{n}=\frac{(-1)^{n}}{2^{n}n!}\frac{d^{n}}{dx^{n}...
4
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1answer
103 views

F-L expansion, $\sum _{n=0}^{\infty } \left(\frac{\binom{2 n}{n}}{4^n}\right)^2\frac{1}{(2 n+1)^4}$, $_pF_q$ and MZV

Background: I'm searching for hypergeometric representations for high level MZVs. I tried several methods to generalize known identities to higher weights and they all succeeded, except the following ...
8
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2answers
114 views

Fourier Legendre expansion for $\frac{\text{Li}_2(x)}{x},\frac{\log ^2(1-x)}{x},\frac{\log (x) \log (1-x)}{x}$

Background: I'm trying to compute some harmonic sums using Fourier Legendre expansion. For instance, the following expansion $$\frac{\log (1-x)}{x}=\sum _{n=0}^{\infty } 2 (-1)^{n-1} (2 n+1) P_n(2 x-1)...
0
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1answer
48 views

Approximating $\sin(x)$ on $[−1, 1]$ in $L^2$ with a second degree polynomial

$p_n : [−1, 1] → \mathbb{R} (n \in \mathbb{N})$ is a polynomial of degree $n$. $p_n$ is an orthonormal system, $\int_{-1}^{1}p_n(x)p_m(x)dx=\delta_{m,n}$. The first task was to calculate $p_0, p_1$ ...
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0answers
19 views

Legendre expansion of the Dirac delta function

There is a known expansion for the Dirac delta function in terms of the Legendre polynomials as $\delta(x) = \sum_{k = 0}^{\infty} (-1)^k \frac{(4k + 1) (2k)!}{2^{2k + 1} (k!)^2} P_{2k}(x)$. I would ...
0
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0answers
25 views

Associated Legendre functions recurrence formula

I have generating function for Associated Legendre functions : $$g(z,t)=(2m-1)!!\frac{(1-z^2)^{m/2}t^m}{(1-2zt+t^2)^{m+1/2}}$$ I need to find generalization of Bonnet's recursion formula (I need to ...
0
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1answer
24 views

Integrating Legendre polynomials

I need to solve following integral: $I_{n}=\int_{-1}^{1}\frac{1}{x}P_{n}(x)P_{n-1}(x)dx$. I have hint that following equation needs to be used: $(n+1)I_{n+1}+nI_{n}=2$. Does anyone have idea how to ...
0
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1answer
38 views

Show that $P_n(x) ={}_2F_1\left(-n,n+1;1;\frac{1-x}{2}\right)$.

I am told that $$P_n(x) ={}_2F_1\left(-n,n+1;1;\tfrac{1-x}{2}\right),$$ where $P_n(x)$ is Legendre polynomial and ${}_2F_1\left(a,b;c;z\right)$ is hypergeometric function. I am just wondering how to ...
0
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1answer
31 views

legendre polynomial prove using recurrence relations

Equation 1: $$P_n(x) = \begin{cases} 1, & \text{ if } n = 0; \\ & \\ x, & \text{ if } n = 1; \\ & \\ \dfrac{1}{n}[(2n-1)xP_{n-1}(x)+(n-1)P_{n-2}(x)], & ...
0
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1answer
36 views

Integral over triple Legendre polynomials involving derivatives

I know the integral over the triple product of Legendre polynomials (see Legendre Polynomials Triple Product), which reads \begin{align} \int_{-1}^{1} P_k(x)\,P_l(x)\, P_m(x) \;\mathrm{d}x = 2 \begin{...
0
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1answer
31 views

Proving Recursive Integral Identity

I am trying to solve the following problem: Given the definition of $I_N(u)$ as \begin{equation} I_{N}(u) \equiv \frac{1}{2\pi} \int^{\infty}_{0} ds \int^{2\pi}_{0} d\phi\,e^{-s} \left(1 + us + 2\...
0
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0answers
30 views

Legendre polynomial formula

Using the formula below to obtain Legendre polynomial $P_2(x),P_3(x) $. $$P_n(x) = \begin{cases} 1, & \text{ if } n = 0; \\ & \\ x, & \text{ if } n = 1; \\ &...
0
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1answer
41 views

Recurrence relations for Legendre polynomials prove by power series

Given that $(1-2tx+t^2)\dfrac{\partial G}{\partial x} - Gt = 0$ and the generating function $G(x;t) = \dfrac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^{\infty}P_n(x)t^n,$ show that $$P'_{n+1} - 2xP'_n + P'_{...
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0answers
8 views

Finding the steady state temperature in a ball using Legendre polynomials

Suppose a ball of radius $1$ is exactly half immersed into ice, so that the bottom half of the surface is at $0^{o}C$, while the upper half is kept at $10^{o}C$. Find the first three nonzero terms in ...
0
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2answers
67 views

Given the Rodrigues' formula for Legendre's polynomials, show that it satisfies the ODE.

The Rodrigues Formula for Legendre's Polynomials is $P_{l}(x)=\frac{1}{2^{l}l!}\frac{d^{l}}{dx^{l}}(x^2-1)^l$. I wrote $P_{l}(x)=\frac{1}{2^{l}l!}\frac{d^l}{dx^l}\sum_{k=0}^l(-1)^{k-l}\frac{l!}{k!(l-...
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0answers
15 views

spherical Bessel - Legendre relation, incident wave

I was reading the partial wave expansion for incident and scattered wave. I cannot understand two things: 1. Why in this process it indicates that the relation indicated in the following picture is ...
0
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0answers
44 views

Legendre Polynomial prove by integration.

Let $f:[-1,1] \rightarrow \rm I\!R$ be defined by $f(x) = \sqrt{\frac{1-x}{2}}$. By multiplying both sides of the equation $\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=1}^{\infty} Pn(x)t^n$ by $f(x)$ and ...
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0answers
34 views

Integral involving Legendre polynomial and exponential function

I am wondering if there is a nice representation of the following integral $$\int_{-1}^1 P_n(x)\,\mathrm{e}^{\,\mathrm{j}\,\lambda\,|\bar{x}-x|}\,\mathrm{d}x$$ with $\bar{x}\in\mathbb{R}$, $\lambda\...
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0answers
18 views

Closed expression for infinite sum of legendre polynomials.

Is there any closed expresion for sums of the type \begin{equation} \sum_{n=1}^{\infty} \frac{P_{n+k}(x)}{n} \end{equation} For some positive integer $k$ ? Also, it would be of much help if there ...
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0answers
15 views

First Derivative of Legendre Polynomials Evaluated at s=1

I want to calculate the first derivative of the k-th Legendre-polynomial evaluated at s = 1 using the generalized product rule$$\frac{d^{n}}{ds^{n}}[F(s)G(s)]=\sum_{j=0}^n {n \choose j}F^{(n-j)}(s)G^{(...
3
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1answer
30 views

First Derivative of Legendre Polynomials

I have been tasked with calculating the first derivative of the k-th Legendre polynomial $$P^{'}_k(1)$$ I was given the hint to use the generalized product rule $$\frac{d^{n}}{ds^{n}}[F(s)G(s)]=\sum_{...
1
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1answer
27 views

Legendre Expansions for Derivatives of Delta Function

Expansions of the delta functions, of following type which are called completeness relations is very useful in many problems in physics. $\delta(x-y) = \sum_{n=0}^\infty P_n(x)P_n(y) \frac{2n+1}{2} $ ...
-1
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1answer
31 views

A question about Legendre polynomials

If $P_n(x)$ is a Legendre polynomial of degree $n$. If a is such that $P_n(a)=0$ that is $a$ is a root of $P_n(x)=0$. Then the $P_{n-1}(a)$ and $P_{n+1}(a)$ is: equal ? or not equal ? Or are of ...
1
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1answer
35 views

Non-analytic smooth functions and Legendre polynomials

From Wikipedia: a function $f$ is real analytic on an open set $D$ in the real line if for any $x_0 \in D$ one can write $$f(x) = \sum_{n = 0} ^ \infty a_n (x - x_0) ^ n$$ in which the coefficients $...
0
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0answers
15 views

How to integrate a single Legendre polynomial $P_l^m(x)$ times a polynomial of $x$

I'd like to be able to find an analytic expression for the integrals: $$ \int_{-1}^{1}(1-x^2)^{m/2} P_l^m(x) dx $$ and $$ \int_{-1}^{1}x^2(1-x^2)^{m/2} P_l^m(x) dx $$ where $P_l^m(x)$ is an ...
1
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0answers
84 views

Associated Legendre Polynomials recurrence relations

I am trying to find the following recurrence relation for these polynomials concerning its derivative: $$(1-x^2)\frac{dP_l^m}{dx}=-lxP_l^m+(l+m)P_{l-1}^m$$ employing the generating function: $$T_m(...
6
votes
1answer
89 views

Definite integral involving Legendre polynomials with weight function $\sqrt{1-x^2}$

While investigating a problem in acoustic scattering in bounded domains, I encountered the following integral: $$\int_{-1}^{1}\frac{\text{P}_n(x)\text{P}_m(x)}{\sqrt{1-x^2}}\mathrm{d}x$$ Where $\text{...
0
votes
1answer
42 views

Laplace Integral for Legendre Polynomial

Using the identity for the Legendre polynomial (where $C$ surrounds z): $$P_n(z) = \frac{1}{2\pi i} \int_C \frac{(\zeta ^2 - 1)^n}{2^n (\zeta - z)^{n + 1}}~d\zeta$$ Prove $$P_n(z) = \frac{1}{\pi} \...
2
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1answer
86 views

Integrating odd Legendre polynomials using generating function

I must show using generating function of Legendre polynomials, that \begin{align} \int_0^1 P_{2n+1}(x)\phantom{1}dx = (-1)^n\frac{(2n)!}{2^{2n+1}n!(n+1)!} \end{align} My attempt is to change the ...
1
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1answer
30 views

Show that $\phi_{n}$ satisfies $((1-x^{2})y')'+n(n+1)y=0$.

Show that $\phi_{n}$ satisfies $((1-x^{2})y')'+n(n+1)y=0$, where $\phi_{n}$ is the Legendre polynomial of degree $n$. Here is what I did so far: $\begin{align*} ((1-x^{2})y')'+n(n+1)y&=0\\ ((1-x^...
1
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1answer
35 views

Differentiate $((x^{2}-1)f_{n-1}(x))^{(n)}$.

Differentiate $((x^{2}-1)f_{n-1}(x))^{(n)}$. Using Leibniz rule, I obtain the following: $\frac{x^{2}-1}{2^{n}n!}f_{n-1}^{(n)}(x)+\frac{x}{2^{n-1}(n-1)!}f_{n-1}^{(n-1)}(x)+\frac{1}{2^{n}(n-2)!}f_{...
0
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0answers
29 views

Monotonicity of Legendre functions (of first kind) with respect to theirs order

I will denote by $P^m_l(x)$ the Legendre first kind function of order $m$ and of degree $l$, $m$ is a positive integer and $l$ is real number. I fix the value of $x\in[0,1]$ and $l=-0.5$. Is there any ...
0
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0answers
46 views

Stieltjes polynomials

Is there a recursive formula to generate the Stieltjes polynomials with respect to the Legendre polynomials or the Legendre function of the 2nd kind? Stieltjes polynomials is $$ E_{n}(x) $$ I've ...
3
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0answers
35 views

Solving PDE with Legendre functions

I'm studying a paper which has a PDE of the form $$\frac{\partial p}{\partial L}(L,n) = A\bigg((n^2-1)\frac{\partial p}{\partial n}(L,n)\bigg),\quad n>1,$$ with a Dirac delta initial condition $p(...
1
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1answer
82 views

Why swapping between the derivative operator and this infinite sum leads to different results?

While working on a mathematical physical problem, i came across seemingly contradictory results. Notations Let's consider $\mathbf{x}_1$ to be the origin of a spherical coordinate system and $\...
0
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0answers
29 views

Legendre polynomials verify $P_n(\cos a) \leq 1$ for any $a$

The following relation needs to be proved: $$|P_n(\cos a)|\leq1,$$ where $a$ is real and $P_n$ is the $n$th Legendre polynomial. I tried using the Rodrigues formula but I'm not being able to ...
2
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0answers
29 views

Decay rate of the coefficients in the Legendre series expansion

Let $u$ be a function in $L^2_{\nu}([0,\pi])$ where $\nu=2\sin(x)$ Let $\hat{u}_n$ denote the truncated Legendre series expansion of $u$ defined by \begin{equation*} \hat{u}_n:= \...
0
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1answer
35 views

legendres polynomial and recurrence formula or rodrigues method

im not being able to do the below sum....tried with recurrence and all other methods not coming please help The question is given below:- P'n+1 +P'n= P0+3P1 +5P2 +.....+(2n+1)Pn where Pn= legendres ...
0
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0answers
23 views

Integral of Legendre Polynomials over half space

I am wondering if there is a way to compute the following integral: $\int_{\frac{\pi}{2}}^{\pi}\left[P_0(\cos\theta)P_k(\cos\theta)\sin(\theta)d\theta\right]$ I have very little experience working ...
0
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1answer
79 views

Why do the Legendre Polynomials have these coefficients?

I learned of the Legendre polynomials for the first time, in the context of finding an orthogonal basis for $\text{span} \{1, x, x^2, ... \}$. According to Wolfram, the Legrndre Polynomials are $$...
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0answers
70 views

Evaluate $\int^1_{-1} \frac{P_{l_1}^{m_1}(x)P_{l_2}^{m_2}(x)}{\sqrt{1-x^2}} dx$

I wish to evaluate $\int^1_{-1} \frac{P_{l_1}^{m_1}(x)P_{l_2}^{m_2}(x)}{\sqrt{1-x^2}} dx$. There is a neat way to show the integral is zero for certain combinations of ms and ls shown here: ...
1
vote
1answer
239 views

Legendre polynomial recurrence relation proof using the generation function

I want to prove the following recurrence relation for Legendre polynomials: $$P'_{n+1}(x) − P'_{n−1}(x) = (2n + 1)P_n(x)$$ Using the generating function for the Legendre polynomials which is, $$(1-...
0
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0answers
27 views

Recurrence relation of Legendre polynomials via generating function [duplicate]

I have to show that recurrence relation of Legendre polynomials posted below is true $$ nP_n(x)= xP_{n}'(x) - P_{n-1}'(x) $$ Hint is to use generating function and differentiate with respect to x ...
1
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1answer
53 views

What non-zero function is w-orthogonal to all the polynomials of degree less than or equal to $n$?

Background It is understood that a function $q$ is $w$ orthogonal to a function $p$ over $[a,b]$ if there holds: $$ \int_{a}^{b} q(x)w(x)p(x)dx = 0$$ For example, for $w(x):=1$, $[a,b]=[-1,1]$, the ...
0
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1answer
152 views

The L2 Norm of Legendre Polynomials

I need help in proving:$$\int_{-1}^{1}P_n^2(x)dx=\frac{2}{2n+1}$$ using the following formula:$$xP'_n-P'_{n-1}=nP_n,\ n=1,2,...$$ (where $P_n$ are the legendre polynomials). Thanks!
0
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1answer
26 views

Conventions used with Legendre Polynomials in spherical harmonics.

I have used Ambisonics audio for 10 years. I have a grasp of the maths on a trigonometric level and have spent the last six months studying Stroud's Advanced Engineering Mathematics and am learning ...
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0answers
36 views

Sum of Legendre Polynomials

I want to prove the following, and would really appreciate if anyone could point me in the right direction. $$P_n(-1/2) = P_0(-1/2)P_{2n}(1/2) + P_1(-1/2)P_{2n-1}(1/2) + \dots + P_{2n}(-1/2)P_0(1/2)$$
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0answers
35 views

Expansion of $ln(\operatorname{cosec} {x}) $ in terms of Spherical harmonics or Legendre Function

I need the coefficients in: $$ln(\operatorname{cosec} {x})= \sum_{0}^{\infty} \sum_{-l}^{l} A_{lm} \mathcal{Y(\theta,\phi)}$$ So, I used here standard technique to multiply both sides by the ...
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0answers
116 views

What's the connection between Legendre polynomials, Associated Legendre polynomials, and Sphereical Harmonic?

Associated Legendre Polynomials and Legendre Polynomials are strongly connected. From what I read, Legendre polynomials is a special case of Associated Legendre polynomials where $m=0$. Thus the ...

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