Spectral::Swsh::SpinWeightedSphericalHarmonic Class Reference

A utility for evaluating a particular spin-weighted spherical harmonic function at arbitrary points. More...

#include <SwshInterpolation.hpp>

Public Member Functions
	SpinWeightedSphericalHarmonic (int spin, size_t l, int m) noexcept
void	evaluate (gsl::not_null< ComplexDataVector * > result, const DataVector &theta, const DataVector &phi, const DataVector &sin_theta_over_2, const DataVector &cos_theta_over_2) const noexcept
ComplexDataVector	evaluate (const DataVector &theta, const DataVector &phi, const DataVector &sin_theta_over_2, const DataVector &cos_theta_over_2) const noexcept
	Return by value the spin-weighted spherical harmonic evaluated at theta and phi. More...
std::complex< double >	evaluate (double theta, double phi) const noexcept
	Return by value the spin-weighted spherical harmonic evaluated at theta and phi.
void	pup (PUP::er &p) noexcept
	Serialization for Charm++.

Detailed Description

A utility for evaluating a particular spin-weighted spherical harmonic function at arbitrary points.

Warning

This should NOT be used for interpolation; such an evaluation strategy is hopelessly slow compared to Clenshaw recurrence strategies. Instead use SwshInterpolator.

Member Function Documentation

evaluate() [1/2]

ComplexDataVector Spectral::Swsh::SpinWeightedSphericalHarmonic::evaluate ( const DataVector &  theta, const DataVector &  phi, const DataVector &  sin_theta_over_2, const DataVector &  cos_theta_over_2  ) const

noexcept

Return by value the spin-weighted spherical harmonic evaluated at theta and phi.

Details

The additional values sin_theta_over_2 and cos_theta_over_2, representing \(\sin(\theta/2)\) and \(\cos(\theta/2)\) are taken as required input to improve the speed of evaluation when called more than once.

evaluate() [2/2]

void Spectral::Swsh::SpinWeightedSphericalHarmonic::evaluate ( gsl::not_null< ComplexDataVector * >  result, const DataVector &  theta, const DataVector &  phi, const DataVector &  sin_theta_over_2, const DataVector &  cos_theta_over_2  ) const

noexcept

Return by pointer the values of the spin-weighted spherical harmonic evaluated at theta and phi.

Details

The additional values sin_theta_over_2 and cos_theta_over_2, representing \(\sin(\theta/2)\) and \(\cos(\theta/2)\) are taken as required input to improve the speed of evaluation when called more than once.

The formula we evaluate (with various prefactors precomputed, cached, and optimized from the factorials) is [50]

\begin{align*} {}_s Y_{l m} = (-1)^m \sqrt{\frac{(l + m)! (l-m)! (2l + 1)} {4 \pi (l + s)! (l - s)!}} \sin^{2 l}(\theta / 2) \sum_{r = 0}^{l - s} {l - s \choose r} {l + s \choose r + s - m} (-1)^{l - r - s} e^{i m \phi} \cot^{2 r + s - m}(\theta / 2). \end{align*}