# 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 ...
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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 ...
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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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### 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 (...
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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 \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 ...
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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 ...
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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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### 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^\...
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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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