Questions tagged [spherical-harmonics]
Questions on spherical harmonics, a set of basis functions that satisfy an orthogonality relation over the sphere.
341
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L_1 norm of spherical harmonics
Let $Y_{k,j}:{\mathbb S}^{n-1}\to {\mathbb R}$ be spherical harmonics on $n-1$ -dimensional sphere.
We know that $\|Y_{k,j}\|_{L_2({\mathbb S}^{n-1})}^2 = \int_{{\mathbb S}^{n-1}}Y^2_{k,j}(x)d\sigma(x)...
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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:
$$\int_{0}^{2\pi} \int_{0}^{\pi} Y_{lm}^2(\theta, \phi)Y_{l'm'}...
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0
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Does the Dirichlet eigenvalue problem make sense on a sphere?
The Dirichlet eigenvalue problem on a given suitable domain $\Omega\subset\mathbb{R}^n$ asks one to find such function(s) $u$ and eigenvalue(s) $\lambda$ that
$$\begin{cases}\Delta u &= \lambda u\...
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products of real spherical harmonics
Based on the description in wikipedia and the book: Modern Quantum Mechanics (Sakurai & Napolitano), any product of two complex spherical harmonics follows the contraction rule:
$$Y_{\ell_1}^{m_1}...
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Poisson Equation for a perturbed sphere - both exterior and interior solutions
I am trying to solve a Heat transfer problem in a slightly ellipsoidal geometry using the Poission Equation, and Cauchy matching conditions on the boundary between the interior (which is a finite ...
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Questions on Gauss's geometric interpretation of spherical functions
(This question was initially posted on HSM stackexchange, after that I came to conclusion that it is too mathematical to be answered there and asked it on mathoverflow. However, it recieved no ...
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Explanation of expanded spherical harmonics?
I’m reading through A Physical Introduction to Suspension Dynamics and I am having trouble understanding the jump between equations $(2.5)$ and $(2.6)$ in the photo. How do the partial derivatives in $...
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When are spherical harmonic expansions valid?
It is known that a square integrable function on the sphere can be expanded in a basis of spherical harmonics,
$$
f(\theta,\phi) = \sum_{l=0}^{\infty} \sum_{m=-l}^l c_l^m Y_l^m(\theta,\phi)
$$
where $\...
- 996
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Seeking help in understanding the proof of the mean value property for harmonic functions
I am currently trying to understand the proof of the mean value property from 'Harmonic Function Theory' by Axler, Bourdon, and Ramet.
Mean-Value Property: If $u$ is harmonic on $\bar{B}(a, r)$, then $...
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Spherical harmonics: sum of three spherical harmonics
Starting from this question How to express multiplication of two spherical harmonics expansions in terms of their coefficients?, I wanted to find the formula for the point-wise multiplication of ...
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How the express the position of a particle using the spherical harmonics?
I have a bit of a weird question, the spherical harmonics are defined as following for $m\ge0$:
$$
Y_{l,m}(\theta,\phi) = (-1)^m \sqrt{ \frac{(2l+1)(l-m)!}{4\pi (l+m)!} } P_{l,m}(\cos(\theta)) e^{im\...
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Why are eigenvectors of Laplace operator commonly used as orthogonal basis?
It seems like, when studying functions defined on some space (e.g. $\mathbb{R} / 2\pi\mathbb{Z}$, $S^2$) and looking for a basis of the space of such functions, the eigenvectors of the Laplacian are ...
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Tensors that describe the symmetry groups
In this article, Table 1 lists different symmetry groups and the tensors that describe them. However, it is unclear how the authors reached these results. They just refer to the book, and it is ...
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Reference for tables of Hankel or spherical Bessel transforms
I am looking for a reference for tables of Hankel/spherical Bessel tranforms. In particular, I'm trying to calculate transforms like
\begin{align}
f_{LM}(r) & = i^L \sqrt{\frac{2}{\pi}} \int_0^\...
- 116
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Linear Combinations of spherical harmonics
The familiar shapes of atomic orbitals arise from spherical harmonics $Y_{\nabla}^{m}$ or their linear combinations. Given:
$Y_1^1 = (\frac{3}{8 \pi})^{1/2} \sin \theta e^{i\phi}$
$Y_1^{-1} = (\frac{3}...
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Are spherical harmonics subject to the Gibbs phenomenon?
I know that a Fourier series of a function that is differentiable will converge to that function. If the function isn't differentiable say it exhibits a jump discontinuity - the Fourier series may ...
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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{...
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Does the Laplace-Beltrami operator on the sphere always commute with the spherical harmonics expansion?
The (real) spherical harmonics $Y^m_\ell$, where where $\ell=0,1,2,\dots$ and $m = -\ell, -\ell+1,\dots,\ell-1,\ell$, are a set of eigenfunctions of the Laplace-Betrami operator $\Delta_{\mathbb S^2}$...
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Convolution on a sphere using generalised fast Fourier transform
I'm reading this paper, which is about spherical convolutions and defining equivariance on a spherical signal. Similar to planar signals where we compute the convolution using FFT by doing $f \ast ψ = ...
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Harmonic function of a sum
Consider the set of three dimensional harmonic polynomials $H_l^m(\vec{x})$ given in spherical coordinates by $H(r,\theta,\phi) = r^lY_l^m(\theta,\phi)$, where $Y_l^m$ is a spherical harmonic.
Now ...
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Is the surface of a sphere 2D or 3D
I'm reading an article and it describes signals on a sphere. I dont understand how SO(3) isn't considered a signal on the sphere. And why is the sphere referred to as S2, when spheres are 3D manifolds?...
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Identify the space of spherical harmonics and the space of homogenous polynomials
Is there an explicit way to identify the space of spherical harmonics and the space of homogeneous polynomials over $\mathbb{C}^2$?
I know that the space of spherical harmonics can be seen the ...
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Spherical Tensors and construction from Cartesian
I have been working on a calculation that involves the gradient of the electric field, specifically using the tensor product of the electric field gradient tensor $\nabla E^* \otimes \nabla E$, and I ...
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hyperspherical harmonics expansion of Gaussian kernel projected to sphere
Suppose we have an $n$-dimensional sphere $S^{n-1}$, defined as $S^{n-1}:=\{x\in \mathbb{R}^n: |x|=1\}$. Let $f:S^{n-1}\to \mathbb{R}{\geq 0}$ be the probability density function of $\frac{X}{|X|}$, ...
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Sampling theorem and interpolation for Spherical Fourier Transform
I am interested in the following transformation:
$$F(\hat r)=\oint_\limits{\text{S}}f(\bar r')e^{i k \bar r ' . \hat r}dS'$$
If $S$ is a sphere with radius $r_0$, ignoring scaling constant $r_0^2$
$$F(...
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Complex conjugate of spherical harmonics
The question asked for square of a complex function. I understand that means the complex function multiplied by its complex conjugate. And that would remove the imaginary part of the complex function ...
- 2,612
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Confusion about spherical harmonics, Legendre polynomials
I'm quite new to the ideas behind spherical harmonics and Legendre polynomials. I have a couple of questions about them.
Spherical harmonics, as I understand them, are functions that can be used to ...
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How to know how to sketch $\cos^2 x$ graphs from $\cos x$?
How to know how to sketch $\cos^2 x$ graphs from $\cos x$?
I am trying to understand spherical harmonics, but I need to brush up on my fundamentals to understand it faster and better.
For example, $\...
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Number of grid points satisfying the triangle inequality
Background: The following questions arise from the Wigner $3j$ symbol, see here. It is well known that the angular momenta $(j_1,j_2,j_3)$ in the Wigner $3j$ symbol must satisfy the triangle ...
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How do the spherical harmonic work
I am trying to understand how the spherical harmonics work, by this I mean, how one plots them and also the square absolute value of it.
Because in Physics we deal with spherical harmonics when we ...
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Can we use zonal spherical harmonics to define the Gegenbauer polynomials?
For a fixed dimension $n$ and degree $k$, let $H$ be the space of all real homogeneous harmonic polynomials of degree $k$ in $n$ variables. We equip $H$ with the inner product $\langle f,g\rangle = \...
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Upper bounds on the expected of product of spherical harmonics on the unit sphere in d dimensions
Say x is a random variable that is drawn from $Unif(\mathbb{S}^{d-1})$, and let $Y_{k,\ell}(x)$ denote the spherical harmonics in $d$ variables. Then are there ways to upper bound the following ...
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Justifying term by term differentiation of spherical harmonics expansion
I saw in many physics texts term by term differentiation of spherical harmonics expansion, but since they're physics texts they're without rigourous proof. Take for example the following from ...
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Radial poisson equation in spherical polar coordinates
(This question may belong on Physics. I put in on Mathematics because I saw more similar questions here.)
I have a seperable density function that is expressed in spherical polar coordinates as:
$$n_{...
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Overlap integral of hyper spherical harmonics
Is there a simple way to compute the overlap integral of three hyperspherical harmonics on the three-sphere?
To be more precise, is there a closed form expression for
$$
\int_{0}^{2\pi}\int_{0}^{\pi}\...
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Problems with the direct solution of Poisson equation in spherical coordinates
I have a charge density $\rho(\vec{r})$ which is given as an expansion of spherical harmonics $Y_{l}^{m}(\hat{r})$.
$$\rho(\vec{r}) = \sum_l^{l_{max}} \sum_{m = -l}^{l} \rho_{l}(r) Y_{l}^{m}(\hat{r}) ...
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Product of two spherical harmonics as a linear combination of spherical harmonics
Studying the book "Physics of Atoms and Molecules" by B.H Brandsden and C.J. Joachain I stumbled upon this given result (without any proof):
$$
Y_{m_1}^{\ell_1}Y_{m_2}^{\ell_2}=\sum_{\ell=|\...
2
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Rotations and Hyperspherical Harmonics
To give a bit of background about my question, let $R$ be a rotation that sends a unit vector $r$ to $r'$ and let $Y_{\ell,m}$ be a spherical harmonic of degree $\ell$ and order $m$ (i.e. $\ell \geq 0$...
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What can be said about multiple derivatives of delta function on the sphere?
For fixed $\beta$ in the sphere $\mathbb S^2\subset \mathbb R^3$, we can define a "zonal derivative" via the distribution $\delta'(\langle \cdot, \beta \rangle)$ on $\mathbb S^2$ -- here $\...
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Power spectrum using Fourier transform on a sphere and spherical harmonics decomposition
Consider an isotropic random gaussian $f(\vec{\theta})$ on the surface of a unit sphere. We can decompose $f(\vec{\theta})$ in 2 ways.
$f(\vec\theta) = \int \frac{d^2 l}{(2\pi)^2} \exp^{i \vec l \...
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Why rotating a function in this paper uses $R^{-1}$ and not $R$?
If you look at Eq. 1 of this (paper)[http://arxiv.org/abs/1801.10130], it states that
"we introduce the rotation operator $L_R$ that takes a function $f$ and produces a rotated function $L_Rf$ by ...
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Orthogonality and completeness of spherical bessel functions
I am interested in computing the following integral, which feels like something that must have been computed before:
$$
\int \frac{k^2{\rm d} k}{2\pi^2}j_{\ell}(r k)j_{\ell'}(r' k)
$$
From what I ...
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Calculation of hessian and gradient of spherical harmonics
I was searching for relations that can be used to calculate the gradient and the hessian matrix of spherical harmonics in Cartesian coordinates easier. I am using the following definition of the ...
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Prove a relation involving the Laplace–Beltrami operator and spherical harmonics
Let $P_ℓ$ denote the space of complex-valued homogeneous polynomials of degree $ℓ$ in $n$ real variables, here considered as functions ${\displaystyle \mathbb {R} ^{n}\to \mathbb {C} }$.
Let $A_ℓ$ ...
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Prove the orthogonal decomposition of the space of spherical harmonics
Let $\mathbb{S}^{n}$ denote the $n$-dimensional unit sphere in $\mathbb{R}^{n+1}$ endowed with the standard metric. For $k=0,1, \cdots$, denote by $\mathscr{H}_{k}^{n}$ the space of spherical ...
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Spherical harmonics orthogonality
I've been struggling with this integral
$$
\int_0^{2\pi}\int_0^{\pi} \sin(\theta) e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'}
(\theta,\phi) \mathrm d\theta \mathrm d\phi
$$
I've tried to use the ...
- 169
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0
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Decomposition into spherical harmonics
I'm trying to follow a text I found online.
The author decomposes EM fields such
$$
\mathbf{E} = \sum_{lm}\left(f_l(r) \mathbf{Y}_{lm} - i \frac{l(l+1)}{r} g_l(r) \mathbf{\Psi}_{lm} - i\left(\frac{d ...
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Approximate $Y_{1,m}$ spherical harmonic with $Y_{00}$.
Given I have the most primitive spherical harmonic, $Y_{00}(\theta, \phi)=\frac{1}{2} \frac{1}{\sqrt{\pi}}$ and I look at one of the three second most primitive ones, e.g. $Y_{11}(\theta, \phi)=-\frac{...
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Calculating the spherical harmonic of θ=π/2
This is a very simple question, yet I'm not sure how to approach it. I want to calculate the spherical harmonic:
$$ Y_{l m}^{*}(\theta = \pi/2, \phi) $$
I know the general formula:
$$ Y_{l m}^{*}(\...
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2
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Do Spherical harmonics have continuous extensions to the entire sphere?
This article contains the following formula for the spherical harmonics:
$$Y_l^m(\theta,\phi) = \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos \theta) e^{im\phi}$$
Now let $S$ be the unit ...
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