# Questions tagged [spherical-harmonics]

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

219 questions
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### Proof for a Legendre polynomial identity on a 2-sphere that appears in cosmology

How can I prove the following identity? \begin{equation} \int d\Omega_{\hat k} P_l(\hat n \cdot \hat k) P_{l'}(\hat n'\cdot \hat k)=\frac{4\pi}{2l+1} P_l(\hat n\cdot \hat n')\delta_{ll'}\label{PP} \...
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### Understanding the Following Integral Notation

I'm a little confused on the notation my professor used for the following integral. \begin{equation} \int \bar{Y}_{l_f}^{m_f} \left( \dfrac{-Y_1^1 + Y_1^{-1}}{\sqrt{2}}, \dfrac{iY_1^1 + iY_1^{-1}}{\...
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### Vector spherical harmonics and resolution

Consider the analysis of a vector function ${\bf F}(\theta,\phi)$ in terms of the vector spherical harmonic (VSH) basis ${\bf Y}_{jm}^{l}(\theta,\phi)$ where $l=j-1,j,j+1$, and $j=0,\ldots, \infty$ ...
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### Is $a^4 + b^4 + c^4$ a spherical harmonic?

I am trying to construct Harmonic polynomials on the sphere. What about the examples: $f = x^2 + y^2 + z^2$ $g = x^4 + y^4 + z^4$ We have that $\nabla^2 f \neq 0$ by compuing the second derivative ...
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### Spherical expansion of an exponential function?

We know that a normal planewave can be Rayleigh expanded by spherical harmonics as$$e^{i\vec k·\vec r}=4π\sum_{l=0}^∞\sum_{m=-l}^li^lj_l(kr)Y_{lm}(\hat{\vec k})Y_{lm}^*(\hat{\vec r}).$$ Does any body ...
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### Does the Addition theorem still holds for real (tesseral) spherical harmonics?

the Addition Theorem state that $$P_{l}(\cos \gamma) = K_l\sum_{m=-l}^{+l} Y_{l}^{m}(\theta, \phi) ~\bar{Y}_{l}^{m}(\theta', \phi')$$ and i have that for real function spherical harminics are ...
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### Expanding a function in spherical coordinates

I have a function f(theta,phi,r) in spherical coordinates. The function dies out at r->infinity (r in my case is dimensionless). Is there a natural way of expanding the function, the same way a ...
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### on the limit of the finite representation of harmonics

Let $Y_n^j, \, -n\leq j \leq n$ be a spherical harmonic. Let $f$ be an even function on the sphere. Then, $f$ can be introduced as $$f=\sum_{j=-n}^A\hat f(j)Y_n^j,$$ where $\sum_{j=-n}^A|a_j|>0$ ...
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### $\Phi(\phi)=A e^{-im \phi}+B e^{im \phi}$ and $\Theta(\theta)=P_l^m(\cos \theta)$ are combined to $P_l^m(\cos \theta) e^{im \phi}$?

$\Phi(\phi)=A e^{-im \phi}+B e^{im \phi}$ and $\Theta(\theta)=P_l^m(\cos \theta)$ are combined to form $$P_l^m(\cos \theta) e^{im \phi}$$ How? Where did $A,B, e^{-im \phi}$ go? Does it read ...
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### Boundary conditions and decomposition on spherical harmonics

Say I have a function $f(x,y,z)$ satisfying some Laplace-type equation on a domain. This domain is an hemisphere with boundary defined by $x=0$ in Cartesian coordinates. I impose either Dirichlet or ...
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### Is there a way to simplify this quotient of factorials?

So I'm implementing numerically this quotient of factorials (in spherical harmonics) and after $\nu=170$ I'm getting an overload error and I was wondering if there was a way to simplify this ...
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### Modeling Lambertian surface by using second order spherical harmonics lighting?

For a 3D surface assumed that its surface appearance follows the Lambertian reflectance i.e. $I = \rho (\mathbb{l} \cdot \mathbb{n})$, I found that it also can be expressed by using following second-...
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### Calculating a normal vector field for a surface defined by spherical harmonics

Given a surface defined by spherical harmonic terms $f(\theta,\phi) = \sum_{l=0}^\infty\sum_{m=-l}^lf_l^mY_l^m(\theta,\phi)$ How can I evaluate the normal vector at each location on the surface? i....
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