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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.

If we wish to apply the Laplacian to a polynomial, we apply it to each term of a homogeneous polynomial (noting that the Laplacian is linear).

A homogeneous harmonic polynomial is a spherical harmonic.

A spherical harmonic is a restriction to the unit sphere of homogeneous harmonic polynomials of degree $n.$

A function on the sphere is harmonic.

You can recover it if you know the spherical harmonics.

Fourier series on the $n-$dimensional sphere are in terms of spherical harmonics.

375 questions
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Spherical harmonics for dummies

Adding to the for dummies. The real spherical harmonics are orthonormal basis functions on the surface of a sphere. I'd like to fully understand that sentence and what it means. Still grappling with Orthonormal basis functions (I believe this is…
bobobobo
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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…
Chris
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Are spherical harmonics harmonic?

According to Wikipedia, a harmonic function is one which satisfies: $$ \nabla^2 f = 0 $$ The spherical harmonics (also according to Wikipedia) satisfy the relation $$ \nabla^2 Y_l^m(\theta,\phi) = -\frac{l(l+1)}{r^2} Y_l^m(\theta,\phi) $$ which is 0…
vibe
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What is the Fourier transform of spherical harmonics?

What is the definition (or some sources) of the Fourier transform of spherical harmonics?
Mack
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How are the "real" spherical harmonics derived?

How were the real spherical harmonics derived? The complex spherical harmonics: $$ Y_l^m( \theta, \phi ) = K_l^m P_l^m( \cos{ \theta } ) e^{im\phi} $$ But the "real" spherical harmonics are given on this wiki page as $$ Y_{lm}…
bobobobo
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An explanation of spherical harmonics?

Could somebody please explain spherical harmonics in a simpler manner than it is demonstrated on various websites (like the Wikipedia page which simply overflows my buffer with symbols). I've tried searching math.SE for some info on it, but it's…
ElcorAllFour
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How to prove completeness of the Spherical Harmonics

Laplace's spherical harmonics "form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions" [1]. I have three related questions about this statement: (1) I can prove their…
okj
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Arranging 8 positive and 8 negative charges to produce $1/r^5$ potential in 3D

Short version of the problem: Given 8 +Q charges and 8 -Q charges in 3D, can I find an arrangement in which their potential has its leading non-zero term proportional to $1/r^5$? Step by step description and motivation? Single +Q charge: In physics…
SSF
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Looking for a reference of integral involving product of four spherical harmonics

We know $$\int d \Omega Y_{l_1m_1}(\theta,\phi) Y_{l_2 m_2}(\theta,\phi) Y_{l_3 m_3 } (\theta,\phi) = \sqrt{ \frac{ (2l_1 + 1)(2 l_2+1)(2l_3+1)}{4\pi} } \pmatrix{ l_1 l_2 l_3 \\ m_1 m_2 m_3 } \pmatrix{ l_1 l_2 l_3 \\ 0 0 0 } $$ Is there any…
user26143
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Spherical harmonics give all the irreducible representations of $SO(3)$?

It is mentioned in Wiki that the spaces $\mathcal{H}_{k}$ of spherical harmonics of degree $k$ give ALL the irreducible representations of $SO(3)$. Could anyone tell me where can I find the proof? Thanks! EDIT: I am seeking for an elementary proof…
Syang Chen
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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} \prod_{j = 2}^{n-1} {}_j…
joy
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Is every harmonic polynomial a linear combination of these?

In $N$-dimensional space, we can show by direct calculation that the polynomial $$ r^{2K+N-2}\nabla_a\nabla_b\nabla_c\cdots \frac{1}{r^{N-2}} \hspace{1cm} \text{(with $K$ derivatives)} $$ is harmonic (annihilated by the Laplacian $\nabla^2$),…
Chiral Anomaly
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Peter-Weyl Theorem on the Sphere

The Peter-Weyl theorem says that the matrix coefficients of the unitary irreps of a compact topological group $G$ form an orthonormal basis for $L^2(G)$. Similarly, spherical harmonics provide an orthonormal basis for $L^2(S^2)$, however the…
Alex Buser
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What is the relationship between spherical harmonics and the schrodinger equation?

spherical harmonics (below image) Schrödinger equation(below image) What is the relationship between spherical harmonics and the schrodinger equation?
User3910
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Rotational invariance

Spherical harmonics functions are said to be "rotationally invariant" On the Wikipedia page, it says: In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary…
bobobobo
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