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

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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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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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44 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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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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23 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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28 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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35 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
71 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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30 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 ...
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80 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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64 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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25 views

a question about Jacobi polynomials

Imagine if I have a defined function $\omega(\alpha, \beta, \gamma)$, where $0<\alpha<2\pi$, $0<\beta<\pi$, and $0<\gamma<2\pi$. I can then expand this function into series just like ...
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1answer
51 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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21 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
37 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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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
109 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
81 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
90 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 ...
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154 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 ...
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1answer
116 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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102 views

Calculating Spherical Harmonic coefficients from discrete experimental data

We have a function defined on the surface of a unit sphere, whose values vary with the polar angle $\theta$ and the azimuthal angle $\phi$, but how they vary is yet to be discovered by an experiment ...
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1answer
175 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
145 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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71 views

A problem on vector calculus on spherical coordinate systems

I'm wondering whether I could get some idea on a problem I've been working on, here it is: h and g are two functions defined on the surface of a unit sphere, and their values depend on the ...
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31 views

Four product of spherical harmonic

I encounter four product of spherical 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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21 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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114 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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43 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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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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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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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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83 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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631 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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64 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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277 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 ...
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$\sum_{m=-l, …,l; l=0,1,2,..} e^{\frac{-i E_l (t_f-t_i)}{\hbar}} Y_{lm}(\phi_f,\theta_f)Y_{lm}(\phi_i, \theta_i)$

I have encountered this series while trying to calculate the path integral of a free particle on a sphere. The sum is $$K=\sum e^{\frac{-i E_l (t_f-t_i)}{\hbar}} Y_{lm}(\phi_f,\theta_f)Y_{lm}(\phi_i, ...
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3answers
2k views

Derivative of Associated Legendre polynomials at $x = \pm 1$

I'm creating meshes for spherical harmonics, and I need a normal at a given point. Whenever I'm at the poles, $\cos{\theta} = \pm 1$, and I do not know how to find the derivative there. All the ...
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3answers
93 views

Klein Bottle discrete harmonics?

Studying discrete representations of a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus. ...
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1answer
701 views

Hartree potential (Coulomb) in spherical symmetry (expansion of spherical harmonics)

I have a question regarding the Hartree potential in spherical symmetry. Specifically, The Hartree potential reads: ...
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1answer
1k views

Spherical harmonics expansion for a particular function

On the unit sphere, each square-integrable function can be expanded as a linear combination of spherical harmonics : $$ f(\theta,\phi) = \Sigma_{l=0}^\infty \Sigma_{m=-l}^{+l} f_{lm} Y_{lm} ...
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Triple Product Integral on Real Spherical Harmonics Basis Functions

Okay I know that Real Spherical Harmonics are given by If $m \lt 0$ $~$ then $\sqrt{2}$ $~$ $Im(\text{SphericalHarmonicY}[l,|m|])$ If $m=0$ $~$ then $~$ $\text{SphericalHarmonicY}[l,0]$ If $m \gt ...
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Variation on a completeness relation

The completeness relation for the spherical harmonics is: $$\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{lm}^*\left(\theta_1,\phi_1\right)Y_{lm}\left(\theta_2,\phi_2\right) = ...
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Common zeros of associated Legendre functions

Suppose that $x_{0}$ is a zero of the associated Legendre function $P_{n}^{m}(x)$ (the degree $n$ is a positive integer while the order $m$ is an integer in the range from $0$ to $n$). If there exist ...
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1answer
1k views

How to prove spherical harmonics are orthogonal

A lot of texts and derivations eg here simply say: "The Spherical Harmonics are orthonormal, so: $$ \int{ Y_l^m Y_{l'}^{m'} } = \delta_{ll'}\delta_{mm'} $$ And if you try any (l,m) pair you will ...
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Lower bound for the eigenvalue

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ ...
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148 views

Hermite functions and integral

Let $$ h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}, $$ where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, be Hermite function. Consider $$ ...
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2answers
884 views

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 ...
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905 views

Is convolution in spherical harmonics equivalent to multiplication in the spatial domain?

Spherical harmonic convolution is defined as: $$ ( k \star f )^l_m = \sqrt{ \frac{ 4 \pi }{2l+1} } h^l_m f^l_m $$ I have a function with RGB values for every $(\theta,\phi)$ in the spatial domain. ...
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193 views

What does $ \langle Y_{lm} | Y _{\lambda\mu} \rangle = \delta_{l\lambda} \delta _{m\mu} $ mean?

In Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion, I can almost completely understand the recurrence relations described, but for one part. The $Y^l_m$ function is ...