Tagged Questions
1
vote
1answer
69 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 :
$$ f(\theta,\phi) = \Sigma_{l=0}^\infty \Sigma_{m=-l}^{+l} f_{lm} Y_{lm} ...
2
votes
0answers
123 views
Common zeros of associated Legendre functions
Suppose that $x_{0}$ is a zero of the associated Legendre function $P_{n}^{m}(x)$ (the degree $n$ is a positive integer while the order $m$ is an integer in the range from $0$ to $n$). If there exist ...
2
votes
0answers
46 views
Lower bound for the eigenvalue
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
...
2
votes
0answers
121 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
$$
...
2
votes
1answer
235 views
How are the “real” spherical harmonics derived?
How were the real spherical harmonics derived?
The complex spherical harmonics:
$$
Y_l^m( \theta, \phi ) = K_l^m P_l^m( \cos{ \theta } ) e^{im\phi}
$$
But the "real" spherical harmonics are given ...
2
votes
1answer
73 views
What does $ \langle Y_{lm} | Y _{\lambda\mu} \rangle = \delta_{l\lambda} \delta _{m\mu} $ mean?
In Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion, I can almost completely understand the recurrence relations described, but for one part.
The $Y^l_m$ function is ...
1
vote
1answer
166 views
How to show integral of different order Hankel transformed functions are equal?
Say I have a function $p_v(r) \in L^2(\mathbb{R})$ given by
$$p_v(r) = \int_0^\infty P(k) J_v(rk)\,k\,dk$$
From mucking around in MATLAB it seems the following is true:
$$\int_{r=0}^\infty ...
4
votes
2answers
494 views
An explanation of spherical harmonics?
Could somebody please explain spherical harmonics in a simpler manner than it is demonstrated on various websites (like the Wikipedia page which simply overflows my buffer with symbols). I've tried ...
0
votes
1answer
79 views
Are there names for the indices of the spherical harmonics?
I know that physicists call $\ell$ and $m$ the "azimuthal" and "magnetic" quantum numbers, respectively. But those sound very physics-y. (I am actually a physicist, but still.) Are there names for ...
2
votes
0answers
339 views
Spherical harmonics give all the irreducible representations of $SO(3)$?
It is mentioned in Wiki that the spaces $\mathcal{H}_{k}$ of spherical harmonics of degree $k$ give ALL the irreducible representations of $SO(3)$. Could anyone tell me where can I find the proof? ...
2
votes
3answers
227 views
What is the integral of three orthonormal basis functions?
The hyperspherical harmonics, given by:
...
