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

learn more… | top users | synonyms

1
vote
1answer
22 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 ...
0
votes
0answers
11 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 ...
0
votes
0answers
25 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 ...
0
votes
0answers
18 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 ...
0
votes
0answers
17 views

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 ...
1
vote
0answers
22 views

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 ...
0
votes
0answers
19 views

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} = ...
0
votes
0answers
15 views

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 ...
0
votes
1answer
55 views

Integrate this Spherical Harmonics Function [closed]

I am interested in the following integral $$\int_0^{2\pi}\int_0^\pi\mathop{\mathrm{d}\theta}\mathop{\mathrm{d}\phi} \sin\theta Y_l^{m*}(\theta,\phi)Y_{l'}^{m'}(\theta, \phi)\cos^2\theta\cos^2\phi,$$ ...
1
vote
0answers
19 views

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: ...
0
votes
0answers
9 views

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 ...
0
votes
0answers
15 views

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? ...
0
votes
0answers
22 views

Rotating the Spherical Harmonics around the x axis

I was trying to rotate the Spherical harmonics around the x axis by an angle of $\pi/2$ radians. At first, I thought that adding a simple substitution like $\theta \rightarrow \theta+\eta$ and $\phi ...
3
votes
1answer
56 views

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 ...
1
vote
1answer
49 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 = ...
0
votes
0answers
44 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 ...
-1
votes
0answers
28 views

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 ...
0
votes
0answers
31 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 ...
3
votes
3answers
119 views

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 ...
1
vote
0answers
14 views

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 $m=a-b$, ...
0
votes
0answers
22 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 ...
1
vote
0answers
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. ...
3
votes
0answers
23 views

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 ...
0
votes
0answers
119 views

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 ...
2
votes
0answers
24 views

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 ...
2
votes
2answers
53 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 ...
0
votes
0answers
17 views

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. ...
0
votes
1answer
40 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, ...
0
votes
0answers
39 views

Closure for Legendre Polynomials

Suppose I have a PDE which can be solved by an expansion on the Legendre Polynomial basis, for an axisymmetric problem in spherical coordinates. For the derivation for such a problem, see these ...
1
vote
1answer
75 views

Projection of a 3D spherical function to a carteasian axis

I have a 3D function defined in a spherical coordinate system $(r,\theta,\phi)$, which is written as a product of a radial function $R_{nl}(r)$ and a spherical harmonic $Y_{lm}(\theta,\phi)$ I.e $$ ...
1
vote
0answers
33 views

Coefficients and synthesis of Associated Legendre Polynomials

First of all, all the Associated Legendre Polynomials (ALP) I'm mentioning below are NORMALISED according to the convention of Spherical Harmonics, and the ALPs can be accessed in Mathematica using ...
1
vote
0answers
27 views

Integrating over particular grids to obtain Spherical Harmonic coefficients

Theoretically the spherical harmonic expansion coefficients of a function $f$ should be calculated via a continuous integration: $$F_{lm} = \int_{0}^{2\pi}\int_{0}^{\pi} ...
1
vote
1answer
41 views

Eigenfunctions of $-\Delta_{S^n}$

I read somewhere that the eigenvalues of the Laplacian $-\Delta_{S^n}$ on the sphere $S^n$ consist of $k^2 + (n - 1)k$, with the corresponding eigenspace $V_k$ consisting of homogeneous harmonic ...
0
votes
1answer
43 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 ...
0
votes
0answers
79 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 ...
3
votes
0answers
41 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}\, :\ ...
3
votes
1answer
340 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 ...
4
votes
0answers
52 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 ...
0
votes
2answers
100 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|}}] & ...
3
votes
0answers
66 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
42 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 ...
6
votes
2answers
160 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 ...
1
vote
2answers
332 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
101 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 ...
1
vote
0answers
49 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 } ...
1
vote
0answers
51 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
104 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
178 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
62 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
163 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 ...