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Questions tagged [spherical-harmonics]

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

If we wish to apply the Laplacian to a polynomial, we apply it to each term of a homogeneous polynomial (noting that the Laplacian is linear).

A homogeneous harmonic polynomial is a spherical harmonic.

A spherical harmonic is a restriction to the unit sphere of homogeneous harmonic polynomials of degree $n.$

A function on the sphere is harmonic.

You can recover it if you know the spherical harmonics.

Fourier series on the $n-$dimensional sphere are in terms of spherical harmonics.

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Calculating the normalization constant in spherical harmonics?

Anyone know a presentation of the calculation of the normalization constant in spherical harmonics. Specifically, how has $$\sqrt{\frac{2l+1}{4 \pi} \frac{(l-m)!}{(l+m)!}}$$ been found in $$Y_l^m(\theta, \phi) = \sqrt{\frac{2l+1}{4 \pi}…
mavavilj
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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? Also since $m1 + m2 + M$ has to be zero for the Wigner…
physicsGuy
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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 possible to write a Spherical Harmonics of a…
drjrm3
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Proper definition of "order" of spherical harmonics?

I'm looking into spherical harmonics, and the term 'order' seems to be used quite inconsistently. In google's spherical harmonics framework (https://github.com/google/spherical-harmonics), the constant spherical harmonic is called order 0 so the…
matthias_buehlmann
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Finding Magnetic Declination

I am trying to find an exact "plug and chug" formula for calculating magnetic declination given time, date, latitude, longitude, elevation, etc. Everywhere I look tells me to use their web based calculator. I was hoping there was a way that I…
Daddy Shark
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eigenfunctions and eigenvalues of multiplication by spherical harmonic

Let $Y_{\ell m}$ be a real spherical harmonic, and define an operator on functions $f:S^2\to\mathbb{R}$ by $$(L_{\ell m}f)(\theta,\phi) = Y_{\ell m}(\theta,\phi)f(\theta,\phi).$$ What are the eigenfunctions of $L_{\ell m}$ (which must be the same…
user7530
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Spherical Harmonic Derivative

This question is a follow up to a previous question: Spherical Harmonic Identity. Instead of using the above question's method, I tried something like this, but don't get the same result and I'm unsure why that is. $$\partial_{\theta_0}…
Karl
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Spherical Harmonic Identity

I've been told there exists the identity $$\partial_\theta Y_{\ell,m}(\theta,\phi)=\frac{1}{2}e^{-i\phi}\sqrt{(\ell-m)(\ell+m+1)}Y_{\ell,m+1}-\frac{1}{2}e^{i\phi}\sqrt{(\ell+m)(\ell-m+1)}Y_{\ell,m-1},$$ but can't quite figure out the proof. If I…
Karl
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Valid spherical harmonics coefficients values

For some cases of spherical functions, like BRDF or a dot product between a sample direction and a surface normal, projecting them to spherical harmonics coefficients gives values over 1 or below -1, e.g. when the dot product is always 1 for any…
Dorota Kadłubowska
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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 try this for a function on a sphere of arbitrary…
Klamauk
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Linear Combinations of spherical harmonics

The familiar shapes of atomic orbitals arise from spherical harmonics $Y_{\nabla}^{m}$ or their linear combinations. Given: $Y_1^1 = (\frac{3}{8 \pi})^{1/2} \sin \theta e^{i\phi}$ $Y_1^{-1} = (\frac{3}{8 \pi})^{1/2} \sin \theta e^{-i\phi}$ Show that…
user307640
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Approximate $Y_{1,m}$ spherical harmonic with $Y_{00}$.

Given I have the most primitive spherical harmonic, $Y_{00}(\theta, \phi)=\frac{1}{2} \frac{1}{\sqrt{\pi}}$ and I look at one of the three second most primitive ones, e.g. $Y_{11}(\theta, \phi)=-\frac{1}{2} \sqrt{\frac{3}{2 \pi}} \sin \theta e^{i…
ste
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Del of spherical harmonics

I am looking for the del of spherical harmonics. For the laplace operator we have the defining relation: $\Delta Y^l_m = l(l+1) Y^l_m$. So, what is with: $\nabla Y^l_m $
nuemlouno
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Legendre Polynomials - representation

Can anybody tell me how I come can rearrange the following formulae into each other: $$\sum_{k=0}^{\frac{l}{2}} (-1)^k {{2l-2k}\choose{l}} {l\choose{k}} x^{l-2k} = \sum_{k=0}^l {l\choose{k}}^2(x-1)^{l-k}(x+1)^k .$$ These are two different…
Philinator
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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 implicitly that one chooses $A,B$ s.t. they and $e^{-im…
mavavilj
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