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

learn more… | top users | synonyms

0
votes
0answers
11 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) = ...
3
votes
0answers
38 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 ...
0
votes
0answers
18 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 ...
1
vote
0answers
10 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 ...
0
votes
0answers
23 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 ...
1
vote
2answers
73 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 ...
1
vote
0answers
49 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 ...
0
votes
0answers
12 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 ...
1
vote
0answers
14 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 } ...
0
votes
0answers
51 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) = ...
1
vote
0answers
33 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 ...
0
votes
0answers
26 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 ...
0
votes
0answers
35 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', ...
0
votes
0answers
49 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 ...
1
vote
1answer
98 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 ...
1
vote
0answers
42 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
votes
0answers
86 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 ...
0
votes
1answer
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 ...
0
votes
0answers
27 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 ...
0
votes
1answer
56 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 ...
0
votes
0answers
22 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 ...
0
votes
1answer
40 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 ...
3
votes
2answers
44 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} + ...
2
votes
1answer
119 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 ...
0
votes
1answer
90 views

Limit with legendre polynomial

I'm stuck on some scattering homework, could anyone help me with this limit involving Legendre polynomials? $$\lim_{\theta\to 0}\frac{P_n^1 (\cos \theta)}{\sin \theta}$$
2
votes
1answer
93 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
votes
2answers
161 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
votes
1answer
117 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 ...
1
vote
1answer
204 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 ...
1
vote
1answer
171 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. ...
1
vote
0answers
23 views

Four product of sphrerical harmonics

I encounter four product of sphrerical harmonic problems and found it has this equation $Y_{l_1}^{m_1}(\theta,\phi)Y_{l_2}^{m_2}(\theta,\phi) =$ ...
1
vote
0answers
115 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} ...
1
vote
1answer
44 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 ...
5
votes
0answers
105 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 ...
1
vote
0answers
57 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 ...
2
votes
0answers
93 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 $d_n(j)$ ...
1
vote
1answer
84 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 ...
3
votes
0answers
707 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. ...
2
votes
0answers
65 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 $f$?
2
votes
0answers
325 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 ...
1
vote
0answers
47 views

$\sum_{m=-l, …,l; l=0,1,2,..} e^{\frac{-i E_l (t_f-t_i)}{\hbar}} Y_{lm}(\phi_f,\theta_f)Y_{lm}(\phi_i, \theta_i)$

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, ...
4
votes
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 ...
2
votes
3answers
97 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. ...
2
votes
1answer
816 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: ...
1
vote
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} ...
1
vote
0answers
48 views

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) = ...
3
votes
0answers
167 views

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 ...
5
votes
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 ...
2
votes
0answers
63 views

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 $$ ...
2
votes
0answers
152 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 $$ ...