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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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Expectation value of $\cos^2\theta$ with Associated Legendre Polynomials as basis

I am trying to solve this integral $$\int_0^\pi P^m_p(\cos\theta) ~~\cos^2\theta ~~P^m_q(\cos\theta)~~\sin\theta d\theta$$ where $P$ is the Associated Legendre polynomial I tried to solve it by the fact that $$\cos^2\theta =…
Amin
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Confusion about the Legendre polynomials

The Legendre equation is given by $$(1-x^2)y''-2xy'+l(l+1)y=0,$$ which has the solution $$y(x)=a_0\Big[1-\frac{l(l+1)}{2!}x^2+\frac{l(l+1)(l-2)(l+3)}{4!}x^4+...\Big]+a_1\Big[x-\frac{(l-1)(l+2)}{3!}x^3+\frac{(l-1)(l+2)(l-3)}{5!}x^5+...$$ We then…
ODP
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Proving a property of Legendre Polynomials

I've poked around Mathematics Stack Exchange for a while, and while I'm sure this is an elementary problem to you guys, I cannot figure this out. I have found similar solved problem prompts on here, but the issue is that I cannot use Rodrigues'…
DUTCHBAT III
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Generating function for Legendre polynomials at x=1

The generating function is given by, $$\phi(x,h)= (1-2hx+h^2)^{-\frac 12}$$ where $|h|\le1$. Why is it that, $$\phi(1,h)= (1-2h+h^2)^{-\frac 12}=\frac 1{1-h}$$ and not, $$\frac 1{h-1} ?$$
jkarimi
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Legendre Polynomials notation

Im looking at the Legendre Polynomials and trying to recreate the first few terms $${d \over dx} \left[ (1-x^2) {d \over dx} P_n(x) \right] + n(n+1)P_n(x) = 0.$$ What exactly is P_n(x)? and could you give an example of how you would produce the…
MathsPro
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legendre polynomial problem

I am trying to express this equation in terms of theta: $$(1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0$$ where $$x=cos\theta$$ I know I can begin with Legendre's equation: $$(1-x^2)\frac{d^2P_n}{dx^2}-2x\frac{dP_n}{dx}+n(n+1)P_n(x)=0$$ For some reason,…
Jackson Hart
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