# 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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### 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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### 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 ...
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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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### 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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### 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. $\ell \geq 0$...
1 vote
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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 $\mathbb {R} ^{n}\to \mathbb {C}$. 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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I'm trying to follow a text I found online. The author decomposes EM fields such $$\mathbf{E} = \sum_{lm}\left(f_l(r) \mathbf{Y}_{lm} - i \frac{l(l+1)}{r} g_l(r) \mathbf{\Psi}_{lm} - i\left(\frac{d ... 0 votes 1 answer 54 views ### Approximate Y_{1,m} spherical harmonic with Y_{00}. Given I have the most primitive spherical harmonic, Y_{00}(\theta, \phi)=\frac{1}{2} \frac{1}{\sqrt{\pi}} and I look at one of the three second most primitive ones, e.g. Y_{11}(\theta, \phi)=-\frac{... 1 vote 1 answer 166 views ### Calculating the spherical harmonic of θ=π/2 This is a very simple question, yet I'm not sure how to approach it. I want to calculate the spherical harmonic:$$ Y_{l m}^{*}(\theta = \pi/2, \phi) $$I know the general formula:$$ Y_{l m}^{*}(\...
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 ...