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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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Stuck understanding proof $\int_{-1}^{1}P_n^2(x)dx=\frac{2}{2n+1}$ of Legendre Polynomials

Skip past proof (i), I only included it because the proof following refers back to it. Below, I'm stuck on proof (ii), and the two (one,two) similar posts I found used Rodrigues' formula and a different formula. The red one below appears to come…
user301105
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Calculating: $I=\int_{-1}^1{(1+x)^{m+n}P_n(x)}dx$ Where $P_n$ is a Legendre Polynomial.

Calculating: $$I=\int_{-1}^1{(1+x)^{m+n}P_n(x)}dx$$ Where $P_n$ is a Legendre Polynomial. My progress: Using Rodrigues formula: $$I=\dfrac{1}{2^n n!} \int_{-1}^1{(1+x)^{m+n}\large(\dfrac{d}{dx}\large)^n(x^2-1)^n}dx$$ A first interation by parts…
Victor F Salazar G
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The Legendre polynomial.

How to show that any polynomial of degree $\le n$ can be written as a linear combination of the $L_j(x)$ for $j = 0,1,2,...,n$ $P_n(x)=a_0L_0(x)+a_1L_1(x)+a_2L_2(x)+\dots+a_nL_n(x)$ ANSWER :: the legendre polynomials of degree ≤ n, are linearly…
Bob
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Finding coefficients, Legendre polynomials.

Say I have a function $f(\theta) = 1 + \cos^2(\theta)$ that can be expressed terms of the Legendre polynomials. When calculating coefficients should I change the Legendre polynomials from $x$ variables to theta variables? e.g. The third Legendre…
skitt1z1
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What's the Legendre of zero?

I know that this question may look like meaningless, but through solving a question, I encountered with this form of Legendre. $ P_{l}(0)$ or a sum over it. You have any idea about it?
Sina
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What does it mean for a solution to be regular (Legendre equation)?

For example, Wikipedia says: When $n$ is an integer, the solution $P_{n}(x)$ that is regular at $x = 1$ is also regular at $x =−1$, and the series for this solution terminates (i.e. it is a polynomial). What is 'regular' in this context?
74hc595
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Determining a Differential equation, related to Legendre

Given that $ P_l(x) $ is a polynomial solution of Legendre's equation, I want to find the differential equation for the function $ W(x) = (1-x^2) \frac{d^2}{dx^2}P_l(x)$ Since Legendre's equation is $(1-x^2)\frac{d^2}{dx^2}P_l(x) - 2x…
Bret Olson
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Legendre polynomial.

How may I express $$V(\theta,\phi) = (Q/a)(\sin(\theta)\cos(\phi) + 0.5\cos^2(\theta))$$ as a Legendre polynomial? I am confused by the presence of $\cos(\phi)$.
Grtv
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legendre solution for non homogenous equation

given the legendre equation $(1-x^2)y'' - 2xy' + by = f(x)$ why can the solution be a series of legendre polynomials $y(x) = \sum_{n=0}^{\infty}a_n P_n(x)$? i thought legndre solves the homogenous version of the equation so wouldn't a series of…
Beast
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Prove that $\int_{0}^{1} P_{l}(x)dx = \dfrac{P_{l-1} (0)}{l + 1}$

I've tried to prove this through the recurrence relation $$ lP_{l}(x) + P'_{l-1}(x) - xP'_{l}(x) = 0 \rightarrow l\int_{0}^{1} P_{l}(x) dx + \int_{0}^{1} P'_{l-1}(x) dx - x \int_{0}^{1} P'_{l}(x)dx = 0 $$ $$ l\int_{0}^{1} P_{l}(x) dx = P_{l-1}(0)…
Carlos Eduardo Staudt
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Convergence of Legendre Polynomials

I am trying to approximate the function $(1-x)^\sqrt{2}$ using Legendre polynomials on the interval $[0,1]$. I have been using $P_n(1-2x)$ as my polynomials as I want all the polynomials to satisfy $P_n(0)=1$. However, when I use the…
JamesVR
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Legendre polynomial integration, basic

I want to evaluate $$\int_0^1x^2P_n(x)\,dx$$ but the only way out i see is by using $$xP_n(x)=\frac{nP_{n-1}+(n+1)P_{n+1}}{2n+1}$$ twice and since i got many $\int_0^1P_l(x)dx$, i use $$P_l(x)=\frac{P'_{n+1}-P'_{n-1}}{2l+1}.$$ In the endm there a…
Vinicius Fuckner Linhares
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Legendre Polynomial Identity

I was looking for a method to do the following integral: $\int^1_{-1}(1-x^2)\frac{dP_m(x)}{dx}P_n(x)~dx$ I know there should be an explicit representation for the result but I am struggling to work it out.
Rescy_
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Legendre polynomial with recurrence relations

Using $P'_{n+1} - P'_{n-1} = (2n + 1)P_n$, show that $P'_{n+1}(x) = \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} (2(n-2k) + 1)P_{n-2k}(x)$ $P_n(x)$ is legendre polynomial. $$P_{n+1}(x) = (2n+1)P_n(x) + P'_{n-1}(x) $$ $$\implies P_{n+1}(x) = (2n+1)P_n(x)…
Joe
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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 proceed?
Biggs Hoson
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