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 Differential equation, n(n-1) or n(n+1)
I am confused regarding the Legendre Differential Equations' coefficients.
In some books its,
$(1-x^2)y''-2xy'+n(n-1)y=0$
and somewhere it is,
$(1-x^2)y''-2xy'+n(n+1)y=0$
what is its correct form?
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Legendre's Polynomial and spherical harmonics
The differential equation that is satisfied by the Legendre's polynomials is:
$$(1-x^2)y'' - 2xy' + \lambda y = 0 (*)$$
I have also been told that the Legendre's polynomial with the parameter $x = \...
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Integrals of Legendre polynomial and a rational function
Is there are analytic expression of the following definite integral?
$$
\int_{-1}^1 x^\alpha (1-x^2)^{\beta} P_l(x) P_m(x) \text{d}x
$$
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Understanding the Dual Emergence of Legendre Polynomials in Differential Equations and Orthogonalization
I am currently examining the mathematical properties of Legendre polynomials and have observed their emergence in two distinct areas: as solutions to a specific class of differential equations (...
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Legendre Polynomial Triple product with different arguments
I'm trying to integrate this: $$f_{jkl}\langle{\hat{a},\hat{b},\hat{c}}\rangle=\frac{1}{4\pi} \int d{\Omega}_n P_j(\hat{a}.\hat{n})P_k(\hat{b}.\hat{n})P_l(\hat{c}.\hat{n})$$ where $\hat{n}$ is a unit ...
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I want to prove the proposition that the absolute value of integral expression must be monotonically decreasing
For arbitrary $r_0$ and $P_l(\text{cos}(\theta))$ be the Legendre polynomials,
$$
E_n=\int_{r0}^\infty \int_0^\pi -\text{sin}^3(\theta)(\left(
\left(\text{cot}(\theta) \sum_{l=0}^n R_l(r) \partial_\...
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Generating Function for Bivariate Legendre Polynomials?
I am aware of the following standard generating function for single-variable Legendre Polynomials:
$$
\sum\limits_{n=0}^{\infty}P_n(x)z^n = \frac{1}{\sqrt{1-2xz+z^2}}
$$
for $x \in \mathbb{R}, z \in \...
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What is the combinatorics meaning of the generating function for Legendre polynomials?
I know the generating function has been a super useful tool when finding the Legendre polynomials (or other special functions), or even used to estimate the static electric potential. In the Physics ...
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Norm of Legendre Polynomials $P_m(x)$
While studying to prove the norm of Legendre polynomials $P_m(x)$ is $\sqrt{\frac{2}{2m+1}}$, I faced $\int_{-1}^{1} [D^m (x^2-1)^m]^2 dx = (2m)! \int_{-1}^{1} (1-x^2)^m dx.$ $D^m$ stands for ...
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Spheroidal eigenvalues with shifted boundary conditions
I was studying the spheroidal differential equation in relation to calculating solutions for fields in a general Kerr background metric and, as far as I can tell, the eigenvalues $\lambda$ that enter ...
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To expand $f(x)=\begin{cases}-1, &-1<x<0\\+1,&0<x<1\end{cases}$ in Legendre polynomial series and obtain formula for expansion coefficients
Expand the step function
$$f(x)=\begin{cases}-1, &-1<x<0\\+1,&0<x<1\end{cases}$$
in a series of Legendre polynomials $P_l(x)$. Obtain an explicit formula for the expansion ...
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Integral of Squared Spherical Harmonics
The following integral comes out of an expression $\langle |Y_{l,m}(\theta, \phi)|^2\rangle$ over a orientation probability distribution:
$$\int_{0}^{2\pi} \int_{0}^{\pi} Y_{lm}^2(\theta, \phi)Y_{l'm'}...
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Prove from the generating function that even index Legendre polynomials are even functions.
At this link: http://www.phy.ohio.edu/~phillips/Mathmethods/Notes/Chapter8.pdf
The author writes that one can prove from the generating function of Legendre polynomials that $P_{2n}$ are all even and $...
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Integral Formula Involving Legendre Polynomials
The following exercise takes the form;
$\int_{0}^{\infty}f\left(\frac{P_{n+1}\left(x\right)}{P_{n}\left(x\right)}\right)\cdot\frac{1}{P_{n}\left(x\right)^{2}}dx=\left(n+1\right)\int_{0}^{\infty}f\left(...
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How to prove this summation equation? [duplicate]
I'm looking for some hints on proving the following (either directly or by induction):
$$
\sum_{k={0}}^{l/2} \frac{(-1)^k(2l-2k)!}{k!(l-k)!(l-2k)!} =2^l
$$
I do know it is actually true from various ...
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