BipolarSphericalHarmonics

BSHCache

Wrapper around monopolar harmonics that are coupled to evaluate the bipolar harmonics. The cache may be constructed using cache.

biposh(SHT, θ1, ϕ1, θ2, ϕ2, j, m, j1, j2)

Evaluate the bipolar harmonic $Y_{jm}^{j_1, j_2}((\theta_1, \phi_1), (\theta_2, \phi_2))$. SHT may be one of SH(), GSH() and VSH(YT(), B()) where YT and B are vector harmonic and basis types offered by VectorSphericalHarmonics.jl.

biposh(SHT, θ1, ϕ1, θ2, ϕ2, j, m, j1j2modes)

Evaluate the bipolar harmonic $Y_{jm}^{j_1, j_2}((\theta_1, \phi_1), (\theta_2, \phi_2))$ for all (j1,j2) in j1j2modes.

biposh(SHT, θ1, ϕ1, θ2, ϕ2, jmcoll::SphericalHarmonicModes.LM, j1j2...)

Evaluate the bipolar harmonics $Y_{jm}^{j_1, j_2}((\theta_1, \phi_1), (\theta_2, \phi_2))$ for all (j,m) in jmcoll, where j1j2 may be either a 2-Tuple (j1,j2) or a collection of 2-Tuples.

biposh!(Y12::AbstractVetor, B::BSHCache, θ1, ϕ1, θ2, ϕ2, j, m, j1, j2)

Evaluate the bipolar harmonic $Y_{jm}^{j_1, j_2}((\theta_1, \phi_1), (\theta_2, \phi_2))$ and store the result in Y12. The cache B determines the type of harmonic evaluated.

biposh_flippoints(SHT, θ1, ϕ1, θ2, ϕ2, j, m, j1, j2)

Evaluate the bipolar harmonics $Y_{jm}^{j_1, j_2}((\theta_1, \phi_1), (\theta_2, \phi_2))$ and $Y_{jm}^{j_1, j_2}((\theta_2, \phi_2), (\theta_1, \phi_1))$ in one pass, utilizing symmetries of the Clebsch-Gordean coefficients.

cache(SHT, T::Type, j12max)

Allocate the arrays required to evaluate the monopolar harmonics for all modes 0 ≤ j1, j2 ≤ j12max. SHT may be one of SH(), GSH(), or VSH(YT(), B()) where YT and B are vector harmonic and basis types offered by VectorSphericalHarmonics.jl. T is a real type that determines the precision of the monopolar harmonics.

kronindex(::GSH, ind1, ind2)

Given the indices of each GSH vector, return the corresponding index of the bipolar spherical harmonic. The indices ind1 and ind2 must lie within -1 and 1.

kronindex(::VSH, ind11, ind12, ind21, ind22)

Given the indices of each VSH matrix, return the corresponding index of the bipolar spherical harmonic. Note that the indices correspond to ℓ - j for the Irreducible vector spherical harmonic $\mathbf{Y}^{\ell}_{jm}(\hat{n})$.

monopolarharmonics!(B::BSHCache, θ1, ϕ1, θ2, ϕ2, j1, j2)

Update B with the monopolar harmonics $Y_{j_1,m_1}(\theta_1, \phi_1)$ and $Y_{j_2,m_2}(\theta_2, \phi_2)$, where $Y$ may either be a scalar or a vector harmonic depending on B.

monopolarharmonics(SHT, θ1, ϕ1, θ2, ϕ2, j1, j2)

Evaluate the monopolar harmonics $Y_{j_1,m_1}(\theta_1, \phi_1)$ and $Y_{j_2,m_2}(\theta_2, \phi_2)$, where $Y$ may either be a scalar or a vector harmonic depending on SHT, which may be one of SH(), GSH(), or VSH(YT(), B()) where YT and B are vector harmonic and basis types offered by VectorSphericalHarmonics.jl.