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

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Integral of the product of three low-order Spherical Harmonics

Given two functions $f(x)$ and $g(x)$ approximated by Spherical Harmonics with the coefficients $c_i^f$ and $c_i^g$, the integral of the product of the two (approximated) functions can be calculated ...
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1answer
25 views

Spherical Harmonic Expansion On Non Unit Sphere

Is there a way of expanding a scalar field defined on a sphere of radius R in the base of spherical harmonic functions? Everywhere I read about expansions on the unit sphere. What changes if one would ...
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22 views

Does anybody know how to actually derive spherical harmonics in a way that is historically accurate and intuitive?

And by "historically accurate", I mean without resorting to techniques of derivation which were developed after the fact or explanations which use the very concept they're trying to explain. The few ...
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11 views

Fourier transform of spherical harmonics divided by $|\vec{x}|^{3}$

I need a formula for Fourier transform of spherical harmonics divided by third order polynomial of the form $$F^{m}_{l}(\vec{x}) = \frac{Y^{m}_{l}(\vec{x})}{|\vec{x}|^{3}}$$ Spherical harmonics are ...
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18 views

Count the number of linearly independent spherical harmonics of fixed degree

For a given integer $q$, I would like to compute the cardinality of the following set of integers: \begin{equation} N(q; \ell)=\left\{m\in\mathbb{Z}, \ell_{q-2}\ldots \ell_2\in\mathbb{Z}_{\ge 0}\, :\ ...
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11 views

Extract Spherical Harmonics from integral

In physics we may find integrals in the format $ I = \int \mathrm{d}b \, b^2 F(b) \mathrm{d}\Omega' \frac{Y_l^m(\theta',\phi')}{|{\bf a} - {\bf b}|^2} $ where the vector ${\bf a}$ has spherical ...
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1answer
114 views

Relation between Hankel transform and Fourier transform

As a physics student, I ran into the following problem. I left out a lot of context, if anything is unclear please ask me. I quote: The statistic that is observable is the angular correlation ...
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31 views

rotation matrix on the 2-sphere

Let $y_i(s): S^2 \mapsto \mathbb{R}$ with $s = (\theta,\varphi)$ for $i =1,\dots,n$ be a set of orthonormal functions on the 2-sphere (more exactly spherical harmonics) and $\beta: \mathbb{R}^3 ...
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32 views

Software to compute spherical harmonics in higher than 3 dimensions (100 or maybe 500 dimensions)?

I have been trying to find an implementation of Spherical harmonics for higher dimensional data but I couldnt find anything in Sage, Mathematica, Matlab. Does anyone have any idea of a standard/fast ...
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45 views

Is the spherical harmonic representation of a 2D field independent of grid?

What I am currently unable to understand is whether the spherical harmonic representation of a 2D field is in any way tied to the nature of the grid on which decomposition/composition is performed. I ...
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2answers
38 views

In the real spherical harmonics, where does the sqrt(2) factor come from?

The real spherical harmonics can be written in terms of the complex spherical harmonics: $$ Y_{\ell m} = \begin{cases} \displaystyle \sqrt{2} \, (-1)^m \, \operatorname{Im}[{Y_\ell^{|m|}}] & ...
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38 views

Associated Legendre polynomial expansion of $\exp(\xi)$

For a project I need to compute the coefficients of the Associated Legendre polynomial expansion of the $\exp$ function. That is I need to find $b_n$ such that $$\exp(x^Ty) = \exp(\xi) = ...
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0answers
52 views

Riemann-Lebesgue Lemma for Spherical Harmonics expansion

Here is my question: A basic result of classical Fourier analysis is that the fourier coefficients of an $L^1$ function must tend to zero (Riemann-Lebesgue Lemma). Is there analogous result to the ...
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0answers
29 views

How can a spherical harmonic have a complex value at $\varphi =0$?

The spherical harmonics form a complete set of the Hilbert space of square integrable functions on the sphere. However, looking at them, I can't see how they could ever be summed to equal a function ...
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23 views

What's a good primer from linear algebra to spherical harmonics?

I need a topic, a primer, that will be able to introduce me to spherical harmonics and how to translate and use them with the usual tools of linear algebra and calculus, namely matrices, polynomials ...
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32 views

Pattern emerging from expansion of sphere in spherical harmonics

Seeing as one can express any function on the sphere in terms of the spherical harmonics, I am interested in what the sphere itself looks like when expanded. To get a function for the sphere, I use ...
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2answers
153 views

Spherical harmonic expansion of a sphere

Seeing as one can expand any function on the sphere in terms of the spherical harmonics, I was thinking it should be possible to express the function for a sphere itself in terms of them. I have ...
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67 views

Expand $\int_{-1}^0 e^{a\cos{\theta}}J_0(b\sin{\theta})\,d\cos{\theta}$ in spherical harmonics.

I want to solve the integral (a probability density function) $$ g(\gamma)=\int_{-1}^0 e^{-f\cos{\theta}\cos{\gamma}}J_0(-if\sin{\theta}\sin{\gamma})\,d\cos{\theta} $$ numerically, everything is ...
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25 views

Spherical harmonics and convolutions on $S^3$

Thinking of Hopf fibration of $S^3$ I got these two questions: Do $S^3$ spherical harmonics have a simpler expression in the Hopf coordinates? In $S^2$ we can convolve only with zonal functions. It ...
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31 views

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 } ...
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55 views

Problem involving spherical harmonics

I want to evaluate a large integral, part of which is $\int\limits_{0}^{2\pi} \int\limits_{0}^{\pi} \sin\theta \frac{u_{\theta}^2}{2} d\theta d\phi$, with $u(\phi,\zeta) = ...
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42 views

Eigenvalues of rotation invariant operators on 2-sphere

Work on $L^2(S^2)$, where $S^2$ is the 2-sphere. Suppose that I have an operator, $T$, that is rotation invariant. That is, $T$ commutes with $R$ for any rotation operator $R$. Suppose furthermore ...
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31 views

Is there addition theorem for Fourier Harmonics?

We know that in spherical harmonic expansion we have addition theorem, and we can expand a function which depends on $x,x'$ and the angle between thesis two vectors $\cos(\theta_{x,x'})$ by spherical ...
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50 views

Summation of Spherical Harmonics derivatives.

The expression below is the addition theorem for spherical harmonics. $\displaystyle \large P_{\ell}(\mathbf{x}.\mathbf{y}) = \dfrac{4\pi}{2\ell + 1}\sum_{m = -\ell}^lY^*_{\ell m}(\theta', ...
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65 views

On the truncation of spherical harmonics

Suppose there is a function $f(\theta,\phi)$ defined on the surface of a sphere, and $\theta$ and $\phi$ are the polar and azimuthal angles respectively. Similar with the fact that a function defined ...
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1answer
130 views

Spherical Harmonics completeness relation

The answer to this question is probably negative, but let me ask it anyway. If I have an expression of the form $$\tag{1}I =\int\!\mathrm{d}\Omega\, Y_1^0Y_l^{*m}(\theta,\phi) \times ...
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0answers
48 views

Proof of $\vec(r) \times \nabla$ in spherical coordinates

My professor claims, that $\vec{r} \times \vec{\nabla} = \vec{e}_{\varphi} \frac{\partial}{\partial \vartheta} - \vec{e}_{\vartheta} \frac{1}{r\sin \vartheta} \frac{\partial}{\partial \varphi}$ in ...
4
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0answers
112 views

Integral of product spherical harmonics

I'm trying to calculate this integral: $$\int_0 ^\pi\int_0^{2\pi} [\sin\theta(\cos\theta \sin \phi + \sin\theta \cos(2\phi))]^2 \sin\theta d\phi d\theta $$ We could compute this integral ...
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1answer
66 views

Have I done something wrong in this integral?

I have showed most of my steps here so I hope that this is easy to follow. I have the integral $$A = C\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}Y^*(\theta, \phi)f(\theta,\phi)sin(\theta) d\theta ...
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1answer
66 views

Interesting special functions identity involving the inner product of real spherical harmonics with a cosecant weight function

In spherical coordinates $\Omega=(\theta,\phi)\in[0,\pi]\otimes[0,2\pi]$, define the inner product $$C_{L_1m_1}^{L_2m_2}:=\left\langle Y_{L_1m_1},\rho,Y_{L_2m_2}\right\rangle=\int ...
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1answer
32 views

Simple expansion for Spherical Harmonics of a difference?

I am working with $Y_{l,m}(\theta-\theta', \phi -\phi')$ and I believe there is a nice way to write that as a product of Spherical Harmonics, but I cannot derive it or find it anywhere. Is it ...
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1answer
42 views

Expansions onto “bases”…?

When we consider expanding functions into fourier series, or taylor series, or onto the spherical harmonics-are these projections onto a basis? Are these bases complete? How can we show this? I know ...
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2answers
58 views

Derivation of the Angular Component of Spherical Harmonics

Every derivation of spherical harmonics seems to tell me that $e^{im\phi}$ is the most obvious solution in the world to $\frac{\partial^2 f}{\partial \phi^2} = -m^2f$. But what about $Ae^{im\phi} + ...
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1answer
132 views

spherical mean of solution of the helmholtz equation

I'm stuck with this problem. Given a domain $\Omega \subset \mathbb{R}^3$ where the function $u$ satisfies: $u_{xx} +u_{yy}+u_{zz} + k^2 u = 0$, I am asked to find the spherical mean over the sphere ...
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1answer
101 views

Limit with legendre polynomial

I'm stuck on some scattering homework, could anyone help me with this limit involving Legendre polynomials? $$\lim_{\theta\to 0}\frac{P_n^1 (\cos \theta)}{\sin \theta}$$
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1answer
98 views

Any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a coclosed $1$-form?

What is meant by saying that any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a "co-closed" $1$-form? [...Since $H^1$ of $S^3$ is trivial it follows that the ...
4
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2answers
176 views

integral of spherical harmonics over cube

The complex solid spherical harmonics can be defined as $$ U_n^m(\boldsymbol{r}) = r^n P_n^m(\cos{\theta}) e^{im\phi}, $$ where $r,\theta,\phi$ are the usual spherical coordinates of ...
4
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1answer
122 views

$\int_{S^{n-1}}\operatorname e^{ix\cdot \omega}\, \operatorname d\omega$

Given $x \in \mathbb R^n$, there exists a simpler expresion for the integral? $$\int_{S^{n-1}}\operatorname e^{ix\cdot \omega}\, \operatorname d\omega$$ where $S^{n-1}$ is the sphere of $\mathbb ...
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1answer
272 views

Spherical Harmonics expansion of a Dirac Delta at the North Pole

I think all the coefficients for the spherical harmonic expansion of a delta function at the north pole should be a constant (presumably 1), but I'm having difficulties calculating them. Could someone ...
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1answer
208 views

Why does the Kronecker Delta get rid of the summation?

I am working with spherical harmonics and the radial equation (part of Laplace's equation in spherical co-ords). The coefficients and equations with I am working with aren't important to my question. ...
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0answers
25 views

Four product of sphrerical harmonics

I encounter four product of sphrerical harmonic problems and found it has this equation $Y_{l_1}^{m_1}(\theta,\phi)Y_{l_2}^{m_2}(\theta,\phi) =$ ...
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124 views

Matrix-valued expansion in spherical harmonics

I am seeking a clever solution to the following problem. Given $$X(\theta,\phi) = exp(-iA(\theta,\phi))\; B\; exp(+iA(\theta,\phi))$$ with the square, Hermitian matrix $A$: $$A(\theta,\phi) = A_{0,0} ...
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1answer
46 views

Spherical harmonic related integral as sequence

Consider the following integral: $2\pi\int\limits_0^\pi \sin(x) \cos^2(x) Y_{l,0}(x) Y_{l+2,0}(x) \mathrm{d}x$ Wherein $Y_{l,0}$ are the spherical harmonics for $m=0$, so they are not dependend ...
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112 views

the spectrum and determinant of the Laplacian on $S^3$

I came across the following statement in a paper: On $S^3$, the eigenvalues of the vector Laplacian on divergenceless vector fields is $(\ell + 1)^2$ with degeneracy $2\ell(\ell+2)$ with $\ell ...
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59 views

Could A Dynamic System Approximate To Spherical Harmonics?

Spherical harmonics describes electron orbitals in the Schrodinger equation. However the possibility remains that electron orbitals are dynamic systems of interacting particles that merely approximate ...
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108 views

expansion of function on spherical harmonics

Let $f :S^{n-1}\longrightarrow R_+$ be a square integrable function. Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$, $j\in Z_+$ and $k=1, \ldots, d_n(j)$, where $d_n(j)$ ...
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1answer
86 views

Question which involves Riesz potential

Let $f :S^{n-1}\longrightarrow R_+$ , $P_{j,k}. j,k \in N$ be a spherical harmonics (http://en.wikipedia.org/wiki/Spherical_harmonics) and $\displaystyle{f\left(\frac{x}{|x|}\right)|x|^{1-\frac ...
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0answers
904 views

Fourier Transform of spherical harmonics

I am trying seeking for definition (or some source) of the Fourier Transform of Spherical Harmonics (see https://en.wikipedia.org/wiki/Spherical_harmonics). Any help will be really appreciated. ...
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72 views

definition of the Fourier transform of function on the sphere

Let $f: S^{n-1}\longrightarrow R^n$ be even continuous function. What is the Fourier transform of $f$?
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390 views

How to express multiplication of two spherical harmonics expansions in terms of their coefficients?

Consider a spherical harmonics expansion/series like this: $$f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)$$ Presumably if we take two functions on ...