Mathematics Stack Exchange
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Questions tagged [legendre-polynomials]

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

The Legendre polynomials $P_n$ are defined to be solutions of Legendre's differential equation

$$\frac{d}{dx} \left[(1 - x^2) \frac{d}{dx} P_n(x)\right] + n(n + 1) P_n(x) = 0$$

Alternatively, by Rodrigues' formula,

$$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \left[(x^2 - 1)^n\right]$$

The Legendre polynomials occur frequently in physics, and in particular in solving Laplace's equation in spherical coordinates.

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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; \\ & \\ \dfrac{1}{n}[(2n-1)xP_{n-1}(x)+(n-1)P_{n-2}(x)], & \text{…
Joe
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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 $$P'_{n+1} - 2xP'_n + P'_{n-1} = P_n$$ \ Since the power series,…
Joe
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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$$\frac{d^{n}}{ds^{n}}[F(s)G(s)]=\sum_{j=0}^n {n \choose j}F^{(n-j)}(s)G^{(j)}(s)$$ I started by using Rodrigues'…
Anonymous
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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 polynomial
mayukh
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Summation of Legendre polynomial series

How do I find the sum $$\sum_{n=0}^{\infty}(-1)^n P_n(x)$$ where $P_n$ are the $n$th order Legendre polynomials? I tried using the generating function but I was not able to arrive at an answer. Any hints appreciated.
rohit_r
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1st derivative of the Legendre Polynomial

I have difficulty in this problem. Show that the 1st derivative of the Legendre polynomials satisfy a self-adjoint differential equation with eigenvalue $\lambda=n(n+1)-2$.
RK Ali
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Rodrigues' formula proof

Consider the polynomials $P_s (x) = \frac{1}{2^ss!}\frac{d^s}{dx^s}[(x^2-1)^s]$. Prove $\int_{-1}^{1}x^kP_s(x)dx = 0$ for all $k \in \mathbb{N}, k
Numox
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Why the zeros of the orthogonal polynomials are symmetric about the origin if the weight function is even?

Why the zeros of the orthogonal polynomials are symmetric about the origin if the weight function is even? t's killing me and every book i've seen is left as an exercise. I'm studying Gauss Legendre. And also i would like to know why then in the…
lmglm
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Legendre - orthogonality related proof

How can I proceed to prove that there are constants $α_0, α_1, ..., α_n$ such that $x^n = α_0P_0(x) + α_1P_1(x) + ... + α_nP_n(x)$ where $P_n$ is legendre polynomial. I guess that this has to do with orthogonality property and the fact that $P_n$'s…
user635977
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How do I turn $\cos(3\theta)$ into a polynomial?

I am solving some spherical symmetry surface charge stuff. A problem gave me a surface charge of $$\sigma(\theta)=k\cos(3\theta).$$ I was having trouble using this to find the constants of my PDE, so I looked at the solutions...…
Tsangares
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Normalizing Factor of the Legendre Polynomials

How do I find the normalizing factor for the Legendre polynomials from the generating function? The generating function is: $$ \frac{1}{\sqrt{1-2xh+h^2}}= \sum^{\infty}_{l=0}P_lh^l. $$ I am doing this from Boas, 12.23.1. There is a hint that I have…
Tsangares
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prove $P_{2n}(0)=(-1)^n\frac{(2n)!}{2^{2n}(n!)^2}$ of Legendre Polynomials.

Use: $P_n(x)=\frac{1}{2^n}\sum_{m=0}^M(-1)^m\frac{(2n-2m)!}{m!(n-m)!(n-2m)!}x^{n-2m}$ where M=n/2 if even, (n-1)/2 if n odd, to prove $P_{2n}(0)=(-1)^n\frac{(2n)!}{2^{2n}(n!)^2}$ and $P_{2n+1}(0)=0$. There's something I'm missing. If x=0, only the…
user301105
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How to proof that $(n+1)P_{n}(x)-(2n+1)x P_{n}(x)+n P_{n-1}(x)=0$?

How to proof that $(n+1)P_{n+1}(x)-(2n+1)x P_{n}(x)+n P_{n-1}(x)=0$ where $P_{n}(x)=\frac{1}{2^{n}n!}\frac{d^{n}}{dx^{n}}(x^{2}-1)^{n}$.
user3879021
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How to proof the following properties about legendre function?

My teacher wrote the following about the legendre function without proof 1) $P^{-m}_{n}(x)=(-1)^{m}\frac{(n-m)!}{(n+m)!}P_{n}^{m}(x)$ 2) $P_{2n}(0)=\frac{(-1)^{m}}{\sqrt{\pi}}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)}$ I search for proof online but…
MrDi
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Express f(x) as a linear combination of Legendre Polynomials

I have the following Legendre Polynomials, $$P_{0}\left(x\right) = 1$$ $$P_{1}\left(x\right) = x$$ $$P_{2}\left(x\right) = \dfrac{1}{2}\left(3x^{2}-1\right)$$ $$P_{3}\left(x\right) = \dfrac{1}{2}\left(5x^{3}-3x\right)$$ $$P_{4}\left(x\right) =…
Sophie Filer
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