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Questions tagged [spherical-harmonics]

Questions on spherical harmonics, a set of basis functions that satisfy an orthogonality relation over the sphere.

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Looking for a general overview of how harmonics on a torus would work

I'm a very early beginner to learning harmonics, and was wondering if someone answer some very general questions I have, since all of the articles I've found online immediately reference advanced ...
Svenn's user avatar
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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}^\...
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Higher Dimensional Spherical Harmonics in Cartesian Form

Are there any tables in the literature or computer software for computing higher dimensional spherical harmonics in Cartesian form, like this Wikipedia article, which lists them for three dimensions. ...
Andrew's user avatar
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The Helmholtz equation for the spherical harmonics with delta functions

In three dimensions, the Green’s function for the Helmholtz equation with a radiating point source $$ (\nabla^{2}+k_{0}^{2})g(\textbf{r},\textbf{r}')=\delta(\textbf{r}-\textbf{r}') $$ is $$ g(\textbf{...
Chris's user avatar
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Need verification of coefficients in Spherical Harmonic

I made up an example which lacks azimuthal symmetry and exists on the boundary and outside of the sphere: $$\nabla^2 u(r,\theta,\phi) = 0; 0<\theta<\pi,0<\phi<2\pi \\ u(a,\theta,\phi) = a^...
Researcher R's user avatar
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How to linearly combine the components of $\ell=2$ Cartesian tensor to ensure that the transformation matrix is orthogonal?

A Cartesian tensor of rank 2 should can be decomposed into 3 irreducible part: $$X=\frac{1}{3}tr(X)I + \frac{1}{2}(X-X^T) + \frac{1}{2}(X+X^T-\frac{2}{3}tr(X)I)$$ and $S=\frac{1}{2}(X+X^T-\frac{2}{3}...
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Any reference about these topics (waves on the sphere)?

At the end of the section 1.9.2 in the book A Panoramic View of Riemannian Geometry, it has the following text about waves on the sphere: "We will not say much now about spherical harmonics ... ...
yo-yos's user avatar
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Sum of Spherical Harmonics and Rotational Invariance

PROBLEM Suppose that $$ \sum_{m=-l}^{l} c_m Y_m^l(\theta, \phi) Y_m^l(\theta', \phi')^* $$ is rotationally invariant, then how can we show that the $c_m$'s must be all equal? ATTEMPT AT A SOLUTION I ...
Matteo Menghini's user avatar
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calculation about the differential cross section of the metastable state in resonance scattering

I met a problem in the book by John R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions. In the question 13.4 (page:258), I need to integrate: $$ \int^{\infty}_0r^2dr|\psi(\...
Hsu Bill's user avatar
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Algebraic approach to spherical harmonics

I am interested in an algebraic approach to the following theorem: Theorem. Consider the sphere $S^{n-1} \subseteq \Bbb{R}^n$ and for each $k=0,1,2,\ldots$, the space $H^k$ consisting of homogeneous ...
Michał Miśkiewicz's user avatar
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Product of 3 or more Spherical Harmonics

The product of two spherical harmonics can be written as the sum of spherical harmonics with coefficients related to Wigner 3j matrices (ref eqn 16) : $$Y_{l_1m_1}(\theta,\phi)Y_{l_2m_2}(\theta,\phi) =...
haricash's user avatar
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How to plot spherical harmonics? [closed]

Let me start by saying that I am only interested in the mathematical aspect of the thing. I would like to plot just for the fun of it the spherical harmonics that are used to plot the electronic ...
Charlie's user avatar
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Sum of a binomial relation confusion

I'm having trouble in understanding how to use the sum in the relation below: $$ {}_sP_{jm}(\cos\theta) = \frac{(j+m)!}{\!\sqrt{(j+s)!(j-s)!}} \biggl(\!\sin{\frac{\theta}{2}}\biggr)^{\!2j} \, \sum_{...
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Spherical multipole moments after flipping an axis

I have an interior spherical multipole expansion (as in Modern Electrodynamics by Andrew Zangwill): $$f(\textbf{r}):=\sum_{l=0}^{\infty} \sum_{m=-l}^{l} B_{lm} r^{l} Y_{lm}^* $$ with spherical ...
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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, \...
pmu2022's user avatar
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Help with an integral involving associated Legendre functions?

In the last few days I've come across this nice integral involving the associated Legendre functions $$ {\large\int_{-1}^{1}} \frac{P_{\ell}^m\left(u\right) P_{\lambda}^{m}\left(u\right)}{\sqrt{1 - u^...
Rafael Benevides's user avatar
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Relationship between Cylindrical and Spherical Hankel Functions via Integration

I am looking for the cylindrical wave expansions of spherical waves. For example, for zeroth order we have the Sommerfeld identity: $\frac{e^{jkr}}{r}=\frac{j}{2}\int_{-\infty}^{\infty}\frac{e^{jk_z|z|...
M Hossain's user avatar
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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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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 (...
Hakan Akgün's user avatar
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If $u : \mathbb{R}^3 \to \mathbb{R}$ is a smooth function satisfying $\sum_{m=-l}^l \langle u, Y_{l,m} \rangle_{S^2}=0$, is it necessarily radial?

Let $Y_{l,m} : \mathbb{S}^2 \to \mathbb{R}$ be the spherical harmonics, as presented in wiki. $u : \mathbb{R}^3 \to \mathbb{R}$ be a smooth function such that \begin{equation} \sum_{m=-l}^l \langle u, ...
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Verification: Eigenfunction expansion using products of bessel and spherical harmonics

Preface This is a can you please check my work post. Maybe these are frowned upon. However, I have been working on this for a couple of days now. I can use some help. Context I am solving a physics ...
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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 ...
Marcosko's user avatar
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Evaluation of integrals of the cross products of vector spherical harmonics

Context I am studying spherical multipole radiation. I am seeking identities pertaining to the same. Given the properties and orthogonality relation of the vector spherical harmonics [1], what do the ...
Michael Levy's user avatar
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Relation between spherical harmonics in $\mathbb{S}^2$ vs circular harmonics in $\mathbb{S}^1$

I'm trying to better understand how a function on the sphere $\mathbb{S}^2$ can be decomposed in terms of spherical harmonics. In particular I have found this notation, given that $f: \mathbb{S}^2 \...
James Arten's user avatar
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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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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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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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3 votes
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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}...
J. Lizy's user avatar
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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 ...
Prakash_S's user avatar
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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 $...
Ryguy266's user avatar
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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 $\...
vibe's user avatar
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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 $...
RiXaTorAgu's user avatar
2 votes
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156 views

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 ...
LPZ's user avatar
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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\...
HitMan01's user avatar
3 votes
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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 ...
Weier's user avatar
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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^\...
kc9jud's user avatar
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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 ...
user2944352's user avatar
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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{...
user58955's user avatar
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2 votes
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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 $\Delta_{\mathbb S^2}$...
Inzinity's user avatar
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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?...
a__ys's user avatar
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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 ...
Korn's user avatar
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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 ...
D. Brown's user avatar
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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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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, $\...
user307640's user avatar
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3 votes
2 answers
319 views

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 ...
Jiaxin Zhong's user avatar
1 vote
1 answer
109 views

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 = \...
Alan C's user avatar
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4 votes
1 answer
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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 ...
Equivalent Triangle's user avatar
1 vote
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141 views

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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