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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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 ...
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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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Show, that there is no alternative basis for $l=1$ on the sphere s.t. the modulus square of each basis vector is indepenedent of $\varphi$ & $\theta$

Given a vector space with the orthonormal basis $b$ being the spherical harmonic functions for $l=1$, i.e. $b=\{Y_{1}^{-1},Y_{1}^{0},Y_{1}^{1}\}$: $$ \begin{aligned} Y_{1}^{-1}(\theta,\varphi) &= {...
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A Radon transform on the sphere

I would be interest in studying an analogous of the Radon transform on the hypersphere. The regular Radon transform is defined for $f\in L^1(\mathbb{R}^d)$ as $$\forall \theta\in S^{d-1}, t\in\mathbb{...
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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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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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Spherical harmonics as orthonormal basis in quantum mechanics

In this article https://mrtrix.readthedocs.io/en/dev/concepts/spherical_harmonics.html the following statement is given: Spherical harmonics are special functions defined on the surface of a sphere. ...
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Spherical Harmonics Relation

I have an expression like $$ (\mathbf{q+K})Y^{m_{\ell}}_{\ell}(\widehat{q+K}) $$ inside of an integral over $q$and it would be a lot easier to perform the integral if I only had spherical harmonics ...
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Spherical harmonics defined on $S^2$ vs defined on $SO(3)$

I'm reading the following paragraph of these lecture notes (pp 35): I'm trying to understand the difference in defining spherical harmonics as functions on $S^2$ or $SO(3)$. From what I've read we ...
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Trying to understand the relationship between Hydrogen atom, spherical harmonics and central field force in quantum mechanics

I have a problem understanding three arguments in quantum mechanic: When we talk about a particle in a central field we have this kind of Hamiltonian: $$H=\frac{p^2}{2m}+V(r)$$ if we use spherical ...
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A question about the laplacian of homogeneous functions.

I'm reading the following document trying to understand the basics of spherical harmonics. Now, I understand that if we deal with homogeneous functions it's convenient to represent them in polar ...
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Harmonic polynomials property

The textbook i am currently reading states that: If $a_1^2+ \cdots + a_n^2 = 0$ and $a_i \in \mathbb{C}$ then the polynomial $$ f(x) = (a_1x_1 + \cdots a_nx_n)^m $$ is a harmonic polynomial and ...
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Spherical harmonics and dimension of SO(n-1) invariant subspace

Let $K \cong SO(n-1)$ the isotropy group for $e_n$. Let $H_m$ be space of homogeneous polynomials, and $Y_m$ the space of spherical harmonics. H_m^k denote the subspace of $K$ invariant polynomials in ...
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Deducing combination of spherical harmonics

Let's say I have a combination of spherical harmonics e.g. $$Y_{\text{total}} = aY_{4}^{2} + bY_{6}^{2} + cY_{8}^{2}$$ Is there a way to find what is the weight of each spherical harmonic via ...
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Spanning set for $L^2(S^2)$?

It is well know that the spherical harmonics form a basis for $H = L^2_{\mathbb{C}}(S^2)$, the square integrable, complex-valued functions on the 2-sphere. My question is if the functions $f_v(x) = e^{...
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Demonstration of inequality between 2 variances expressions

Just to remind, $C_\ell$ is the variance of random variables $a_{\ell m}$ following a centered Gaussian PDF (in spherical harmonics of Legendre) : $$C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\...
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Spherical Laplacian

I am trying to prove this: For $f \in C^2(S^{n-1})$ that satisfies $\Delta_{S^{n-1}} f = \lambda f,$ then $\lambda \leq 0.$ And if $g \in C^2(S^{n-1})$ satisfies $\Delta_{S^{n-1}} = \mu g$ with $\...
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Taylor series for $\frac{1}{(1-x)^{n}}$

So I am trying to find the dimension of the subspace of homogeneous polynomials of degree m $P_m$ of $n$ variables. For $\alpha = (\alpha_1, \dots, \alpha_n$), with $\alpha_1 +\cdots + \alpha_n = m.$ ...
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Fourier Expansion vs. Real Spherical Harmonics Expansion

This is a weirdly elementary question related to the history of real-valued Fourier expansions for which I cannot find any good references. For simplicity, I am following Wikipedia's conventions with ...
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Spherical Harmonics - Inner product

If $\mathcal{P}$ is space of polynomials in $n$ variables with complex coefficients. Let $\mathcal{P_m}$ be the subspace of homogeneous polynomials of degree m. How would i show that for two ...
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3 votes
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Spherical harmonics - Computing the variance of Poisson noise integrated over $\ell$ on a defined quantity?

It is an astrophysics context but actually, it is mostly a mathematics issue. From spherical harmonics with Legendre deccomposition, I have the following definition of the standard deviation of a $C_\...
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Plotting Spherical Harmonics - Phase Coloring?

Looking at the wikipedia table of spherical harmonics, I'm trying to figure out how they are computing phase and using it set the hue on the plots. I can't figure this out for either the real plots (...
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How do I write $\sin(2𝜙)$ as an expansion of spherical harmonics?

I've been given a wave function, $\psi = f(r)\sin(2𝜙)$, and told to rewrite it as a summation of spherical harmonics. All of the spherical harmonics seem to be written with a $\theta$ dependence so I'...
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SE(3)-equivariant transformations: Wigner-D matrices and spherical harmonics

I am trying to understand the maths behind rotation-equivariant neural networks, for example SE(3)-Transformers, Tensor Field Networks or Steerable CNNs. All these papers talk about decomposing the ...
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Express $\sin(\theta)$ in terms of spherical harmonics

So we can express trigonometric quantities in terms of Spherical harmonics, for example $\cos(\theta)\propto Y^{0}_{1}(\theta,\phi)$. Is there a closed expression for $\sin(\theta)$? If not, is there ...
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Extract a potential part of a function

Given a function $f: \mathbb{R}^3 \rightarrow \mathbb{R}$, expressed in spherical coordinates, suppose I want to decompose $f$ into a potential piece and a non-potential piece, $f = f_P + f_{NP}$ and ...
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Substitution of the imaginary part in Laplace Spherical Harmonics

Let's say we want to solve the integral $$\iint\sin^2(\theta)\sin(\phi) Y_{lm}^*(\theta,\phi)\,d\theta\, d\phi.\tag{1}$$ I have seen in some contexts that you can make the substitution $$\sin(\theta)\...
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8 votes
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How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials

From numerical experiments in Mathematica, I have found the following expression for the integral: $$ \int_{-1}^{1}h_{n}^{(1)}\left(\sqrt{a^{2}+b^{2}+2ab\tau}\right)P_{n}^{m}\left(\frac{a\tau+b}{\sqrt{...
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How can I modify this recurrence relation for the Normalized Associated Legendre Polynomial to use the full normalization instead of spherical?

I'm implementing an algorithm for a numerically stable normalized associated Legendre polynomial, but I need a different normalization factor than the source I'm using. The source is here and uses a ...
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3 votes
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Confusion regarding Wigner D-matrices and rotation of spherical harmonics

I am having trouble understanding the role of Wigner D-matrix coefficients when considering the effect of rotations on spherical harmonics. Suppose that we want to understand the effect of a rotation $...
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What spherical functions can the N-th order limited spherical harmonics represent?

Background It is known that the spherical harmonics $Y_n^m$ with order $n$ and degree $m$ (such that $n \ge 0, -n \le m \le n$) are functions on the sphere which form a complete, orthogonal infinite ...
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Is there any spherical rectangle that is closed under intersection/complement?

In Euclidean space, the rectangles are closed under intersection but not closed under negation/complement. I was wondering whether we can define a rectangle in an unit sphere such that the spherical ...
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3 answers
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Symmetries of a spherical harmonic basis

I am interested in constructing a set of functions $Z_l^m$ which are linear combinations of spherical harmonics up to a maximum degree $L$, $$ Z_k^n(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l \alpha_{...
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Laplacian kernel decomposition over Gegenbauer polynomial

I want to decompose the Laplacian kernel $K(x,y)=e^{-|x-y|_2}$, with $x,y \in \mathbb{R^d}$, over the Gegenbauer polynomials of degree k: $Q_k^{(d)}(\vec{x}\cdot \vec{y})$, which are linked with the ...
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2 votes
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Singularity issue when evaluating first derivative of Spherical Harmonics

I have been trying to use the 3D fast multipole expansion, $$ \Phi(P) = \sum_{n=0}^{\infty} \sum_{m=-n}^{n} \frac{M_n^m}{r^{n+1}}Y_n^m (\theta, \phi) $$ to approximate the source potential in the far ...
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1 vote
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Orthogonality of spherical harmonics under a rotation

Consider the orthonormalized spherical harmonics defined with respect to Cartesian axes $(x,y,z)$ which obey the relation, $$ \int d\Omega Y_l^m Y_{l'}^{m'*} = \int_0^{2\pi} d\phi \int_0^{\pi} d\theta ...
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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 \...
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Find a basis of the vector space of all harmonic homogeneous $n$-variate complex polynomials of degree $d$

Consider $U_n = \mathbb{C}[x_1,...,x_n]$ - the vector space of all complex polynomials over with $n$ variables $x_1,...,x_n$ ($n \ge 2$). A polynomial $h \in U_n$ is homogeneous of degree $d \in \...
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How to calculate Associated Legendre Functions as a function of theta?

I’m trying to solve Problem 4.4 in Griffiths Quantum Mechanics, and I need to calculate the associated legendre functions $P^{m}_{\ell}(x)$ to do this. $P^{m}_{\ell}(x)$ are functions of $x$, yet in ...
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What is meant by a defining representation?

I'm aware that there are similar questions, but none of them are really something that I can approach with my current knowledge of groups and representations (which is the first 3 chapters of a book ...
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How to define the shapes and orientations of orbitals mathematically in a cartesian coordinate system?

I am not able to match the $d$ and $f$ orbitals in atomic structure worksheet given as homework (see the attached image). I was able to match the $p_{x}$ , $p_{y}$ , and $p_{z}$ orbitals. But I am ...
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Hyperspherical and Spectral Decompositions of a correlation matrix

It is difficult for me to understand the following example problem where we have to do a spectral and hyperspherical decomposition of a matrix which is given by enter image description here I know ...
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Need help with this proof from Axler's Harmonic Function Theory

A free, legal copy is available here. On pp. 81-82, I have two questions: Why do the Cauchy-Riemann equations imply that all complex derivatives except the $m^{th}$ vanish at the origin? If anything, ...
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What is the correct formula for $ n $-dimensional spherical harmonics?

In the Wikipedia article, the formula for $ n $-dimensional spherical harmonics is given as $$ Y_{\ell_1, ..., \ell_{n-1}}( \theta_1, \dots \theta_{n-1} ) = \frac{1}{\sqrt{2\pi}} e^{i \ell_1 \theta_1} ...
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Do the spatial spherical harmonics form a complete basis over ${\rm I\!R}^3$?

It is well-known that the spherical harmonics form a complete orthogonal basis over the unit sphere $S^2$. But what about the spatial spherical harmonics (with the additional $r^n$ part), do they form ...
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solution of laplace equation in 3 dimension by speration of variables covers all possible solutions?

The usual procedure to solve the Laplace equation in 3 dimensional space is to switch to spherical coordinates, use the seperation of variables approach, which eventually leads to the well-known ...
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Constructing new spherical basis functions

I am trying to construct a new set of basis functions on the sphere. The reason is I am working in a special coordinate system $(r,\theta_q,\phi_q)$ where $\theta_q = \theta_q(r,\theta,\phi)$ and $\...
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