# 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 Coefficients and more for this question

I' struggling to understand these questions, I'm not sure how to find Legendre coefficients set out like this for part a) and d) just confuses me. I know this isn't a site to ask for help on questions ...
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### How to derive general formula for $\int_0^1P_l(cos\theta)d(cos\theta)$

how do I derive general expression for this Legendre integrals on the path $(0,1)$ $$\int_0^1P_l(cos\theta)d(cos\theta)$$ P.S Of course we can subtitude $cos\theta=x$ it doesn't really matter
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### Differentiation of a product involving divided differences

In 'Collocation at Gaussian Points' by DeBoer (top of p601): We have a function $h_{t}(u) = G(t, u)r[\tau_{1}, ..., \tau_{k}, u]$ being considered on an interval $[a, b]$ where: $G(t, u)$ is a Green's ...
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### Proving that Legendre polynomials decreasing about $n$ as $x$ approaches to 1.

I am in the study of the Legendre polynomials and think about proving $$P_n(x)>P_{n+1}(x)$$ when $x$ is very close to 1. This is obvious when I see this picture on Wikipedia. I was wonder if there ...
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### Alternative definition of Legendre polynomials

I'm studying Panofsky and Phillips' Classical Electricity and Magnetism. In writing the potential of a linear $2^n$-pole lying along the $x$-axis, they make use of the following definition for the ...
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### How to simplify a partial sum obtained by Legendre polynomials

Context I am working on an electrostatics problem. I have undertaken Fourier analysis. By $k$, I denote a natural number $k=0,1,2,\ldots$. I have obtained the following partial sum in terms of the ...
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### Shifted Legendre polynomials symmetry relation

I have to prove that $p_n(1-x)=(-1)^np_n(x)$ ($x\in[0,1]$) for all $n\in\mathbb{N}$, where $(p_n)_n$ is the family of Legendre polynomials on $[0,1]$: given $(x^n)_{n=0,1,\ldots}$, $(p_n)_n$ is the ...
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### Expand Associated Legendre Polynomials in the basis of shifted Associated Legendre Polynomials

I'm tasked to solve the following integral: $$\int_0^{\pi/2}P^m_n(cos(\theta))P^m_l(cos(\theta))sin(\theta)d\theta$$ where $n,k\in Z^+$ and $m \in Z^{0+}$. Is it possible and wise to perform the ...
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### 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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### The coefficient and asymptotic in generating function

Let $L_n$ be the set of all the paths from $(0,0)$ to $(n,0)$ such that every step is $u=(1,1)$ , $d=(1,-1)$ and $r=(2,0)$. Notice that the path could go under the $x$ axis. a. Write a generating ...
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### Integral form of legendre polynomial of second kind

I found a definite integral form of Legendre polynomial of second kind, $$Q_{n}(z)=\frac{1}{2}\int^{+1}_{-1}\frac{P_{n}(t)}{z-t}dt$$ when n is an integer. I wonder how to evaluate this integral. I ...
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### 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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### Orthogonality legendre polynomials when $m = n$.

Now for the result when $n = m$, we use Rodrigues' Formula Theorem. Indeed, \begin{equation}\label{eq:Pn2} \int_{-1}^{1}(P_n)^2dx = \dfrac{1}{(n!)^22^{2n}}\underbrace {\int_{-1}^{1}\dfrac{d^n}{dx^n}(...