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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 problem (please help!)

Problem: Show that $P^{'}_{2n+1}(0)= \frac{(-1)^n (2n+1){^{2n}}C_n}{2^n}$. My attempt: Given: $P_{n}(x)=\frac{1}{2^n}\sum_{k=0}^{\frac{1}{2}n}(-1)^{k} {{^n}}C_k{^{2n-2k}C_n}x^{n-2k}$ Let $n = 2n+1$. We have…
Lawrence Orewa
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How to calculate Integral of sine raised to the power of $2l+1$

I encountered the following integral when dealing with Legendre polynomials, trying to derive their orthonormality relation starting from the Rodriguez Formula. $$\int_0^{\pi/2}\sin(\theta)^{2l+1}d\theta$$ According to my book, the result…
Nick Heumann
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Asymptotic behavior of local maximum/minimum points of the Legendre polynomial

Asymptotically, for the degree $n \rightarrow \infty$, is there an approximation of local extreme points of the Legendre polynomial $P_n(x)$? I am particularly interested in the first (either decreasing or increasing order) local extreme point. I…
Yip
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Derive Jackson Equation 3.26

I want to derive equation 3.26 from jackson's book, classical electrodynamics. $(2l+1)\int_{0}^{1}P_l(x)dx=(-\frac{1}{2})^{(l-1)/2}\dfrac{(2l+1)(l-2)!!}{2(\dfrac{l+1}{2})!}$ where l is odd, using the Rodrigues formula $P_l(x)=\dfrac{l}{2^l…
Otv
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Derivation of Legendre differential equation from Rodrigues formula.

I have to derive corresponding differential equation for Legendre polynomial from Rodrigues formula: $$ P_n(x)=\frac{1}{2^n n!}\left(\frac{d}{dx}\right)^n(x^2-1)^n $$ The solution is indeed $(x^2-1)y''+2xy'-n(n+1)y=0$, but I am not sure, how to…
Vid
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Other properties of Legendre Polynomials

Given the fact: (Asymptotic form) For $\theta \in (0, \pi)$, we have $$ P_n(\cos \theta)= J_0(n\theta)+ O(\dfrac{1}{n})= \sqrt{\dfrac{2}{\pi n \sin \theta}}\sin \left(\dfrac{2n+1}{2}\theta + \dfrac{\pi}{4}\right) + O(\dfrac{1}{n}),$$ (where $J_0$ is…
Joe
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legendre polynomial prove using recurrence relations

Equation 1: $$P_n(x) = \begin{cases} 1, & \text{ if } n = 0; \\ & \\ x, & \text{ if } n = 1; \\ & \\ \dfrac{1}{n}[(2n-1)xP_{n-1}(x)+(n-1)P_{n-2}(x)], & \text{ if } n \geq 2. \end{cases}$$ Equation 2: $$P'_{n+1} - 2xP'_n…
Joe
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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!
Katrine
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On Legendre's Polynomial

I want to show that the coefficient of $x^n$ in $P_n(x)$ is $(2n)!/ (2^n(n!)^2)$ my problem is that I cannot find the the $n$-th derivative of $(x^2-1)^n$ to be able to simplify Rodrigues' formula of pn! can someone give me a hint?
user635977
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Legendre polynomials

I'm relatively new here( This is my first question tbh) This question is from my assignment which I'm supposed to submit in two days. I have been scratching my head over past couple of days but have reached nowhere. Obtain the first three terms in…
Yash Chugh
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Integration of Legendre Polynomials with different arguments

I'm trying to calculate this: $$\sum_{l=0}^{\infty} \int_{\Omega} d\theta' d\phi' \cos{\theta'} \sin{\theta'} P_l (\cos{\gamma})$$ where $P_{l}$ are the Legendre polynomials, $\Omega$ is the surface of a sphere of radius $R$, and $$ \cos{\gamma} =…
alansammarone
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Integral of derivative of Legendre polynomials

Show that: $$\int_{-1}^{1}(1-x^2)P_l'(x)P_m'(x)dx=\frac{2l(l+1)}{2l+1}\delta_{lm}.$$ So here's what I have: $$[(1-x^2)P_n'(x)]'=-n(n+1)P_n(x)$$ (from the original DiffEq), and the orthogonality relation…
J. Hunt
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Legendre Polynomial relation

Show that $$\Sigma_{r=0}^{n}(2r+1)P_r(x)=P_{n+1}'(x)+P_n'(x).$$ I've started with the relation $$P_l'(x)=\Sigma_{r=0}^{\frac{1}{2}(l-1)}(2l-4r-1)P_{l-2r-1}(x).$$ I tried adding the sums for $P_n'(x)$ and $P_{n+1}'(x)$ to show that this is equal to…
J. Hunt
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Shifted Legendre Polynomial

We know that the Legendre Polynomial is $P_n(x) = \frac{1}{2^nn!}\frac{d^n}{dx^n}[(x^2-1)^n]$. The Shifted Legendre Polynomial $\tilde P_n(x)$ is defined as $P_n(2x-1)$. Can you please tell me how to get $\tilde P_n(x) =…
geniusacamel
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Determine an integral with legendre polynomials

Determine the following integral $\int_{-1}^{1}x^2P_{2n-1}(x)dx$ I am unsure how to go about this. I understand that $P_{2n-1}=\frac{1}{2^{2n-1}(2n-1)!}\frac{d^{2n-1}}{dx^{2n-1}}(x^2-1)^{2n-1}$ and…
Lauren Bathers
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