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Tagged with legendre-polynomials spherical-harmonics
32
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Legendre addition theorem in $2$ dimensions
We know the addition theorem for Legendre polynomials in spherical coordinates is
$$P_\ell(\cos\gamma)=\dfrac{4\pi}{2\ell +1}\sum_{m=-\ell}^\ell\mathrm Y_{\ell m}(\theta_1,\phi_1)\,\mathrm Y_{\ell m}^\...
- 2,668
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Expansion of $\frac{\text{erfc}({|\vec{r} - \vec{r'}|})}{|\vec{r} - \vec{r'}|}$ in spherical harmonics?
How can I derive the spherical harmonic expansion coefficients for the function $$ \frac{\text{erfc}({|\mathbf{r} - \mathbf{r'}|})}{{|\mathbf{r} - \mathbf{r'}|}} $$ by expressing it as $$f(\theta, \...
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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 = \...
- 376
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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 (...
- 139
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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 ...
- 175
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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:
$$\int_{0}^{2\pi} \int_{0}^{\pi} Y_{lm}^2(\theta, \phi)Y_{l'm'}...
- 19
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partial alternating sum of legendre polynomial
My question concerns the (generalized) Legendre polynomials in the $d$-dimensional space (see, e.g. Mueller, Freeden & Gutting):
$$
P_n(d;t) = \frac{|S^{d-3}|}{|S^{d-2}|} \int_{-1}^1 (t + i \sqrt{...
- 1,179
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Calculating the spherical harmonic of θ=π/2
This is a very simple question, yet I'm not sure how to approach it. I want to calculate the spherical harmonic:
$$ Y_{l m}^{*}(\theta = \pi/2, \phi) $$
I know the general formula:
$$ Y_{l m}^{*}(\...
- 253
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How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials
By experimenting in Mathematica, I have found the following expression for the integral:
$$
\int_{b-a}^{b+a}\sigma h_{n}^{(1)}(\sigma)P_{n}^{m}\left(\frac{\sigma^{2}-a^{2}+b^{2}}{2b\sigma}\right)P_{n'}...
- 469
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319
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Understanding the dimensionality of Legendre polynomials.
I have been looking at the Laplace equation $\nabla^2 f = 0$ in various dimensions. In 3 dimensions, the angular equation leads to the well-known spherical harmonics, defined up to normalisation as
\...
- 241
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496
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How to calculate the integral of a Legendre polynomial
I would like to show that
$$
\int_{0}^{1}P_{l}(1-2u^{2})e^{2i\alpha u}du=i\alpha j_{l}(\alpha)h_{l}(\alpha)
$$
where $P_{l}(x)$ are the Legendre polynomials, $\alpha$ is a positive constant and $j_{l}$...
- 469
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172
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Spherical Harmonics Sum Identity
I'm taking a course in Quantum Mechanics and this problem is causing me some struggles. Can someone help me prove this identity?
$$\sum_{m = -l}^l m^2 |Y_{l}^{m}(\theta, \phi)|^2 = \frac{l(l+1)(2l+1)}{...
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Parity of spherical harmonics
I would like to prove that $Y_{\ell m}(-\mathbf{r}) = (-1)^\ell\, Y_{\ell m}(\mathbf{r})$. In this formula, $Y_{\ell m}$ are the spherical harmonics given by
\begin{equation}
Y_{\ell m}(\theta, \...
- 456
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Legendre Expansions for Derivatives of Delta Function
Expansions of the delta functions, of following type which are called completeness relations is very useful in many problems in physics.
$\delta(x-y) = \sum_{n=0}^\infty P_n(x)P_n(y) \frac{2n+1}{2} $
...
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