0
votes
1answer
42 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 ...
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} ...
3
votes
0answers
160 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
60 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
146 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
2answers
803 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
168 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
255 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 ...
6
votes
2answers
1k 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
104 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 ...
4
votes
1answer
598 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
304 views