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

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Calculating the normalization constant in spherical harmonics?

Anyone know a presentation of the calculation of the normalization constant in spherical harmonics. Specifically, how has $$\sqrt{\frac{2l+1}{4 \pi} \frac{(l-m)!}{(l+m)!}}$$ been found in $$Y_l^m(\...
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Expressing spherical harmonics as a combination of other spherical harmonics

Spherical harmonics are a useful tool in physics, particularly in classic electrostatics and electrodynamics. Given an integer $l$, the spherical harmonic $Y_{l,m}$, where $-l\leq m\leq l$, solves the ...
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Spherical harmonics: how's Laplace's equation related to spheres?

Many spherical harmonics derivations start from finding a solution to Laplace's equation and the results are in fact what are called spherical harmonics. However, how's Laplace's equation really ...
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9 views

When is a linear combination of spherical harmonics positive?

Let's say I have a real function $f$ on the sphere, and I express it as a linear combination of (real) spherical harmonics: $$f(x) = \sum_{l,m} \alpha_{l,m} Y_{l,m}(x).$$ Are there necessary and ...
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Green's function for the Laplace or Helmholtz equations for an open rectangle in 2d or cylinder 3d?

I have only ever worked with free space Green's functions, or Green's functions for for the upper half space in 2d. So is it possible to determine a Green's function for the Helmholtz equation or ...
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20 views

Spherical Wave Approximation

Lets suppose i have $ K $ data points $(r_i,\phi_i,\theta_i,p_i)$ and i want to approximate my data points with the following function: $$p(r,\theta,\phi) = \sum_{n=0}^{N} \sum_{m=-n}^{n} c_{n,m}h_n^{...
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14 views

Spherical function dot product after rotation

Let's start from a system $f_n(x) = e^{inx}$. I can rotate one element: $f_m(x + x_0) = e^{imx_0} e^{imx}$ , which is still orthogonal to all but one element of system. Same holds for $n$-...
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41 views

When expanding a function in a sphere, why is the complex conjugate of the spherical harmonic function used to calculate the coefficients?

When expanding a function on a sphere $f(r,\theta, \phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}=A_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta,\phi)$. Since what I'm asking involves the ...
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Spherical functions which are invariant under a finite rotation group

Is there a nice, clean reference which lists a basis in terms of (linear combinations of) spherical harmonics for the $L^2$ space of functions defined on the sphere $\mathbb{S}^2$ which are invariant ...
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Rotationally invariant function in three-sphere

I know that every rotationally invariant function $f(x,y,z)$ in two-sphere $S^2$ must satisfy \begin{align*} f(x,y,z)=f(h(\theta)(x,y,z)^T) \end{align*} for all $\theta$ and $x,y,z$, where \begin{...
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28 views

Spherical Harmonic integration

How do I solve the lower integration of spherical harmonics, in terms of Clebsch Gordon coefficients? \begin{equation} \int d\Omega Y^{l_1}_{m_1}(\hat n)Y^{l_2}_{m_2}(\hat n){Y^{l}_m}^*(\hat n) \end{...
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Equality about integral on unit sphere, involving Lebesgue measure

I met an equality in the book "Harmonic analysis and approximation on the unit sphere" by Wang Kunyang, in page 23. I cannot follow the reasoning below, \begin{align} \int_{\Omega_{n}} P^{n}_{k}(\xi\...
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Proof of an inequality in spherical harmonics using Minkowski's inequality

I met an inequality in the book "Harmonic analysis and approximation on the unit sphere" by Wang Kunyang and Li Luoqing. I need some hints to follow the proof. The proof is following, \begin{align} \...
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18 views

An expository reference on spherical harmonics on $S^n$.

I am looking for a thorough reference which explains how to compute the spherical harmonics on $S^n$ and how to upper and lower-bound their values. About the first part of my query, I am not so much ...
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34 views

Are linearly independent harmonic polynomials orthogonal upon integration over the sphere?

There is a theorem that states that the vector space of homogeneous polynomials decomposes into an orthogonal direct sum of vector spaces of harmonic polynomials as $$ \mathcal{P}_n = \mathcal{H}_n \...
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show that $\sum P(x,y) e^{x^2 + y^2}$ is a modular form over $\Gamma_0(4)$ where $P(x,y) = x^4 - 6 x^2 y^2 + y^4$

In these lecture notes of Zagier, I read that generalized theta functions are still modular forms. Let $q = e^{2\pi i z}$ $$\theta(z) = \sum_{(x,y) \in \mathbb{Z}} \Big[ x^4 - 6 x^2 y^2 + y^4 \Big] \...
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Spherical Harmonic fitting excess polar magnitudes

I am trying to fit an expansion of spherical harmonic functions to a dataset distributed over the surface of a sphere using the least squares method. Each data point is in terms of (r,θ,φ) where r is ...
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35 views

How to prove the given property of spherical harmonics?

How to prove: $$ \int^\pi_0\int^{2\pi}_0 Y_{l'',m''_l}(\theta,\phi) Y_{l',m'_l}(\theta,\phi) Y_{l,m_l}(\theta,\phi) \sin\theta \,d\theta \,d\phi = 0 $$ unless $l, l',$ and $l''$ are integers that can ...
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What do these variables mean in regard to the wave equation and spherical waves?

https://en.wikipedia.org/wiki/Wave_equation#Spherical_waves Before it states ''where K=w/c'', there is an equation that has the following variables: d,r,w,c,l. It also has f_lm(r) What do each of ...
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Composite Spherical Harmonics expansion and error propagation

Let's assume that $f$, $g$ and $t$ are three functions defined over the surface of a sphere. In particular $f$ is defined using $g$, $t$ and the integral operator as follows: $$f(\omega)=\int_{\...
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18 views

Are spherical harmonics a basis for $H^1$?

We know that spherical harmonics are a complete orthonormal system for $L^2(\mathbb{S}^2)$. Is it true that they are also a complete orthonormal system for $H^1(\mathbb{S}^2)$? Furthermore, is it ...
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Eigenfunctions of the Dirichlet Laplacian in balls

I am trying to find out about the Dirichlet eigenvalues and eigenfunctions of the Laplacian on $B(0, 1) \subset \mathbb{R}^n$. As pointed out in this MSE post, one needs to use polar coordinates, ...
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How to construct a complete set in $L^2(\mathbb{R}^3)$ starting with the Spherical Harmonics?

The Spherical Harmonics form a complete set of functions on the sphere $S^2$, so that any function of $f: S^2\to \mathbb{R}$ can be written uniquely as $$f(\theta,\phi)=\sum_{l=0}^\infty \sum_{m=-l}^{...
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39 views

Orthogonal functions and different function arguments

This should be a fairly trivial question, but I would like to support my current intuition. Let's consider two orthogonal functions $f$ and $g$, i.e. two function whose inner product is equal to zero ...
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26 views

Closed-form of spherical expansion of Legendre polynomial $P_k(\sin{\theta}\cos{\varphi})$

During the times of working on some problem in astro/geophysics I have come across a problem involving an expansion into spherical harmonic functions (this is the remnance of nomenclature there used ...
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28 views

Laplacian on the sphere (proof of a proposition)

$$ Let \;f:U \to \mathbb{R}\;be\;a\;function\;on\;an\;open\;subset\;of\;S^{m-1}\;which\;is\;the\;restriction\;of\;a\;smooth\;function\;F:U{'} \to \mathbb{R}\;defined\;on\;an\;open\;subset\;of\;{{\...
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26 views

A recommendation for introductory to spherical harmonics

My Background: So I'm familiar with multivariable calculus, Fourier series, ODE, a little bit of PDE, and a bit of linear algebra. Yesterday I read about Legendre polynomials and their properties(...
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Prolate spheroidal coordinates and spheroidal harmonics expansion

I recently started to study problems with prolate spheroidal geometries, for which prolate spheroidal coordinates are most suited. In particular I have the advantage that the problem is axisymmetric ...
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Limit as r tends towards the radial center of a spherical quantum well

Suppose $$R\left ( r \right )=\frac{u\left ( r \right )}{r}$$ Where $$u\left ( r \right )=A\sin\left ( kr \right )+B\cos\left ( kr \right )$$ The boundary condition is $$u\left ( a \right )=0 \wedge ...
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Isotropic random fields on a sphere

Given the spherical harmonic expansion of an isotropic random field $f$ defined on the sphere, $f(\mathbf{n}) = \sum_{l = 0}^\infty \sum_{m = -l}^{l} f_{lm} Y_{lm}(\mathbf{n})$, where $\mathbf{n} = (...
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Spherical harmonics expansions of the delta function

I'm trying to find the derivation of the spherical harmonics expansions of the spherical Dirac Delta function $\delta_{S^2}$. In general, a spherical function $f(\eta)$ can be expanded into spherical ...
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Integrate this Spherical Harmonics Function [closed]

I am interested in the following integral $$\int_0^{2\pi}\int_0^\pi\mathop{\mathrm{d}\theta}\mathop{\mathrm{d}\phi} \sin\theta Y_l^{m*}(\theta,\phi)Y_{l'}^{m'}(\theta, \phi)\cos^2\theta\cos^2\phi,$$ ...
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Spherical resolvent kernel on $H^n(\mathbb R)$

Is there an explicit formula in the literature for the spherical resolvent kernel $R_{\lambda}(r)$ of the Laplacian $\Delta_{H^n}$ on $H^n(\mathbb R)$ the real hyperbolic space ? Such that: $rad(\...
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Resolvent kernel of Hyperbolic space

The expression of the spherical functions $\varphi_\lambda(x)$ and the resolvent kernel $r_\lambda(x)$ on Hyperbolic space are known. The spherical functions are the Jacobi functions $\varphi_\lambda(...
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Product of two spherical harmonics

According to wikipedia, the product of two spherical harmonics is this equation What I do not understand is the scope of the sum. From where to where does it go, over all integers for L and M? ...
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Symmetry planes in spherical harmonic basis

Suppose I have a function $f(x):S^2\rightarrow\mathbb{C}$ in the degree four spherical harmonic basis: $$f(\theta,\varphi):=\sum_{k=-4}^4a_kY_4^k(\theta,\varphi).$$ I have two related questions: Is ...
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57 views

Numerical integration using Gaussian quadrature, but with data at some nodes not available?

In order to calculate the integration of a function $f(x)$ from $[-1,1]$ using Gaussian quadrature, the values of $f(x)$ at the Gaussian nodes (in my case it's for a 6-node calculation, $f(x)$ at $x = ...
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51 views

Spherical Harmonics's sumation identity proof

Could anybody help me proving this identity? I've tried using the completeness property of the spherical harmonics, expanding the right side using the Binomial theorem and the left side as a ...
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Properties of spherical harmonics [closed]

How Are $Y_{lm}$ and $Y^m_l$ related? How are $Y_{lm}^*$ and $Y_{lm}$ related? I am very confused seeing different notations used at different places. Also, some references use |m| except in $\exp(im\...
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34 views

Spherical Fourier Transform of $p_t$

Let $p_t$ denote the heat kernel of the Laplacian $\Delta = \sum_{i=1}^{n}\frac{d^2}{dx_{i}^2}$ on $\mathbb R^n$, I want compute the Spherical Fourier Transform of $p_t$ on $\mathbb R^n$ ? Thank you ...
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I don't know what this symbol means

I somehow made it to grad school without coming across this symbol: $\left( \begin{array}{ccc} l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{array}\right) $ Here, $l_i$ and $m_i$ are all ...
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Show that coefficient of homogeneous polynomial are determined by their degree and the difference of their index.

I am given the homogeneous polynomial of degree $l$: \begin{align} u(x,y,z)=\sum_{a,b}c_{ab}(x+iy)^a(x-iy)^bz^{l-a-b} \end{align} Where $0 \leq a,b \leq l$. I have to show that given $l$ and $m=a-b$, ...
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28 views

Writing Identities Involving the Real Spherical Harmonics Instead of Their Complex Counterparts.

The real spherical harmonics $Y_{\ell m}(\hat{\mathbf{r}})$ can be defined from the complex spherical harmonics, $Y_{\ell}^{m}(\hat{\mathbf{r}})$ as $$ Y_{\ell m} = \begin{cases} \displaystyle {i \...
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30 views

Are the Spherical harmonics the S^2 equivalent of the exp(i \pi n) function series?

As I understand it, the Spherical harmonics and the "Fourier functions" $\exp(i\pi n)$ with $n\in\mathbb{N}$ have much in common: Both are eigenfunctions of the angle part of the Laplace operator. ...
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What are the modes of vibration of a genus-2 surface?

So it's spherical harmonics for a sphere. The vibrations of a torus presumably are just ordinary string harmonics around each loop. But what are the harmonics on a genus-2 surface (a donut with 2 ...
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Rearranging a spherical harmonics expansion

Referring to this article (click to enlarge): and How is it that they get from equation 2 to equation 3? Whenever I do it, I can't cancel the imaginary terms Is there some spherical harmonics ...
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Applying rotation invariant linear operators to spherical harmonics

In the article "On boundary condition for multidimensional diffusion processes" A Venttsel says: I can't see how one can "prove that any other harmonic of order $n$ may be represented as a linear ...
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2answers
55 views

Separation of variables and complex numbers

I began with the Laplace's equation in the context of spherical harmonics. From wikipedia, one reads. So far I have followed, but in the sequel is stated that $m \in \Bbb{R}$ since $\Phi$ is ...
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If I have two Spherical harmonic series equal to one another, are the coefficients the same?

Let's say I have the Spherical harmonic series decomposition of a tensor with given series coefficients, and it is equal to another spherical harmonic series decomposition with unknown coefficients. ...
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1answer
46 views

Laplace equation in spherical coordinates

I am trying to calculate the Laplace equation($\Delta f =\partial_x\partial_xf + \partial_y \partial_y f + \partial_z \partial_z f = 0$ ) in $\Bbb{R}^3$ for spherical coordinates. $$g(r, \phi,\theta)...