Questions tagged [legendre-functions]

This tag is for questions relating to Legendre Functions (or Legendre Polynomials), solutions of Legendre's differential equation (generalized or not) with non-integer parameters.

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43 views

How can I derive the Legendre function of first kind in terms of the hypergeometric function?

I was reading in Wikipedia about Legendre's differential equation. I was particularly interested in the simple case of the equation given by $$ \left(1-x^2\right)y'' -2xy' + \lambda(\lambda+1)y = 0 \...
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0answers
22 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,...
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1answer
110 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\\ \...
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20 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 ...
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0answers
28 views

How to find the associated Legendre functions

Reading the Courant-Hilbert Methods of Mathematical Physics (p.326) we encounter: "...If we differentiate equation $$ \left[(1-x^2)u')\right]'+\lambda u=0 $$ with respect to $x$, we obtain a ...
2
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0answers
63 views

Is this task impossible?

I'm dealing with what I can only assume is an impossible problem. I want to find the coefficients such that $$x^2+2y^2+3z^2=\sum_{l=0}^\infty \sum_{m=-l}^l a_{lm}\rho^l P_l^m (cos(\theta))e^{im\phi} |...
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23 views

A relation on Legendre functions with real index

Consider the function $f$ defined for $j \in ]0 , 1[$ by $$f(j) = P_{1-\gamma}\bigg( \frac{1}{j}\bigg) - \frac{1}{j} P_{2-\gamma}\bigg( \frac{1}{j}\bigg) $$ where $P_{\alpha}$ is the Legendre function ...
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26 views

How to prove this relationship using properties of Legendre polynomials?

I was asked to prove this relation in my assignment. Any solutions for this??
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1answer
25 views

Finding the Gauss-Legendre quadrature over an interval

So i have been stuck at this problem for a while.. Does anyone have some hints to how it can be solved? Or maybe show me how its done? Find the Gauss-Legendre quadrature over the interval ...
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18 views

Convergence of a serie to obtain the eigenvalues of general Legendre operator

We have the Legendre operator, $$ \mathcal{L}u = - \frac{d}{dx}\bigg[ (1-x^2)\frac{du}{dx}\bigg], $$ and we want to find its eigenvalues and eigenfunctions, therefore we must solve the following ODE, $...
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32 views

Differentiation of Rodrigues's Formula for Legendre polynomial

How to differentiate the Rodrigues's Formula for Legendre polynomial which is given as follows $$P_n(x)=\frac{1}{2^n.n!}\frac{d^n}{dx^n}(x^2-1)^n$$ I could not proceed a single step in this. I know it ...
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0answers
49 views

Legendre Polynomials proving a relation using Rodrigue's Formula

I have been trying to prove the relation$\left (P'_n(1)=\frac{1}{2}n(n+1)\right )$ and have proved it directly from the Legendre differential equation as $P_n(x)${the Legendre polynomial} is the ...
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25 views

Derivation of the relation between Associated Legendre Function and Gauss Hypergeometric Function

I was trying to derive the Hobson's relation between the associated Legendre function and Gauss Hypergeometric function. This is given by, $$P^m_n(z)=\frac{1}{\Gamma(1-m)}\left(\frac{z+1}{z-1}\right)^{...
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1answer
80 views

Showing a summation identity for $1$, possibly tied to Legendre polynomials

The Problem: Consider the sign function on $(-1,0)\cup(0,1)$ defined by $$ \sigma(x) := \left. \text{sgn}(x) \right|_{(-1,0)\cup(0,1)} = \begin{cases} 1 & x \in (0,1) \\ -1 & x \in (-1,0) \end{...
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1answer
22 views

Does Associated Legendre Function of Second Kind Give Delta Function?

Associated Legendre Function of Second Kind is singular at $x=\pm 1$. So I am wondering whether it satisfies the corresponding differential equation everywhere or there is a hidden functional of delta ...
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0answers
24 views

Integral of Legendre's first kind function of degree $-1/2+il$

Let $l=-1/2+i\,t$; $t\in\mathbb{R}$. I'm wondering if there is a way to compute the following integral $$\int_{-1}^xP^m_l(\gamma)^2 \,d\,\gamma; -1<x<1.$$ In which $P^m_l$ is the Legendre's ...
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1answer
103 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}...
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19 views

Evaluation of associated Legendre function of the first kind with complex arguments

I am trying to evaluate in Matlab an expression involving the associated Legendre function of the first kind $P_\nu^\mu(z)$. If my understanding is correct, this function has two slightly different ...
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24 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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1answer
142 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 ...
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0answers
42 views

Associated Legendre Function of Second Kind

The associated Legendre Function of Second kind is related to the Legendre Function of Second kind as such: $$ Q_{n}^m(z)= (-1)^m (1-z^2)^{m/2} \frac{d^m}{dz^m}(Q_{n}(z)) $$ The recurrence relations ...
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0answers
42 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 ...
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1answer
30 views

How to prove that Spherical Harmonics must have integer order $m$ and degree $n$?

Spherical harmonics : \begin{equation} Y_{m}^{n}(\theta,\phi) = N_{}\mathcal{P}_{m}^{n}(\cos \phi) e^{in\theta}, \end{equation} are the solution of the two angular equations (Legendre associated eq ...
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1answer
86 views

Convex envelope of a function

I am trying to calculate the convex envelope of a function, which is calculated by doing the Legendre transform twice (right?). Therefore, I was trying to calculare the convex envelope of $f(x)=(x^2-1)...
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1answer
36 views

By changing the variable in the φ equation to x = cos φ, derive the self adjoint form of the Legendre equation

We have $$\frac{d}{d\phi}(sin\phi \frac{dP}{d\phi}) + \lambda sin\phi P =0 $$ By changing the variable in the equation to x = cos $\phi$, derive the self adjoint form of the Legendre equation: $$\...
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1answer
45 views

Can't retrieve Legendre equation from Laplace equation

I am tasked with the above, converting the 2-dimensional polar form of the Laplace transform: $$ \frac{\partial}{\partial r}\Bigl(r^2 \frac{\partial z}{\partial r}\Bigr) + \frac{1}{\sin(\phi)} \frac{...
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1answer
54 views

derivative of normalized associated Legendre function at the limits of x = +/-1

I'm trying to compute the derivative of the normalized associated Legendre function of $x=\cos \theta$ and I'm having issues when $\cos \theta =\pm 1$ which causes the denominator to go to $0$. I've ...
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0answers
36 views

find $a_L$ in $ \frac{Y_{\ell}^{m}Y_{\ell'}^{-m}}{\sin^2\theta}=\sum_{L=0}^{\ell+\ell'}a_LY_L^0$

Can you prove that $$ \frac{Y_{\ell}^{m}Y_{\ell'}^{-m}}{\sin^2\theta}=\sum_{L=0}^{\ell+\ell'}a_LY_L^0~, $$ where $Y_{\ell}^{m}$ denotes the spherical harmonics of degree $\ell$ and order $m$ (both are ...
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1answer
83 views

How to convert a hypergeom function to the Legendre function?

Anyone can help me to convert the following maple pdsolve expressed by the hypergeom function to the $LegendreP(n,b,x)$ or $Q$ function? \begin{equation} dsolve\Big( (1-x^2)\cdot \frac{d^2 y(x)}{dx^...
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1answer
50 views

Transforming an ODE into Legendre's Equation

I am trying to transform the ODE $$\frac{1}{\sin(\theta)}\frac{d}{d\theta}\left(\sin(\theta)\frac{dS}{d\theta}\right)+\lambda S=0,$$ in to Legendre's equation $$(1-\mu^2)\frac{d^2S}{d\mu^2}-2\mu\frac{...
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1answer
167 views

Integrating multiple products of order 1 Legendre functions

I would like a closed form expression for the integrals of some products like the following: $$ \int_{-1}^1 P^1_j P^1_k P^1_l P^1_m dx $$ and $$ \int_{-1}^1 P^0_j P^0_k P^1_l P^1_m dx, $$ where $P^...
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0answers
69 views

An upper bound of the first eigenvalue of Laplacian on a Riemannian manifold.

I'm reading the Cheng's thesis ""Eigenvalue Comparison Theorems and Its Geometric Applications," and the author obtains an estimate of eigenvalues of the Laplacian based upon his theorem: If $M$ is $...
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1answer
210 views

Series Legendre Polynomial

We have to prove $$\ln \Big(\frac{x+1}{1-x}\Big)=\sum_{n≥0} \frac{x^{n+1}}{n+1}P_n(x)$$ using the generatrix function of Legendre polynoms. I don't know if it is useful, but $$\int_{-1}^{1}\frac{1}{...
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0answers
33 views

Equivalence of spherical harmonic definitions

In Griffith's "Introduction to quantum mechanics", the spherical harmonics are defined for integers $l$ and $-l\leq m\leq l$ as $$Y_{l}^m(\theta,\phi) = \epsilon \sqrt{\frac{2l+1}{4\pi}\frac{(l-|m|)!}...
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1answer
335 views

Sum of associated Legendre functions

I want to find the sums of two expressions involving the Schmidt-normalized associated Legendre functions. They are defined by \begin{align} S_l^0(x) &= P_l^0(x) \\ S_l^m(x) &= \sqrt{2 \frac{(...
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1answer
209 views

Legendre Polynomial Of Second Kind

How to calculate the normalisation factors of Legendre Polynomial of second kind? It is provided that ,the normalisation factors are chosen so that second kind Polynomials satisfies the recurrence ...