Header SphericalHarmonics.h

Detailed Description

Library of Spin-weighted Spherical Harmonic functions and an implementation of the Wigner-D matrices.

Author

S.Fairhurst, B. Krishnan, L.Santamaria, C. Robinson, C. Pankow

Prototypes
COMPLEX16	XLALSpinWeightedSphericalHarmonic (REAL8 theta, REAL8 phi, int s, int l, int m)
	Computes the (s)Y(l,m) spin-weighted spherical harmonic. More...
int	XLALScalarSphericalHarmonic (COMPLEX16 *y, UINT4 l, INT4 m, REAL8 theta, REAL8 phi)
	Computes the scalar spherical harmonic \( Y_{lm}(\theta, \phi) \). More...
INT4	XLALSphHarm (COMPLEX16 *out, UINT4 L, INT4 M, REAL4 theta, REAL4 phi)
	Computes the spin 2 weighted spherical harmonic. More...
double	XLALJacobiPolynomial (int n, int alpha, int beta, double x)
	Computes the n-th Jacobi polynomial for polynomial weights alpha and beta. More...
double	XLALWignerdMatrix (int l, int mp, int m, double beta)
COMPLEX16	XLALWignerDMatrix (int l, int mp, int m, double alpha, double beta, double gam)
	Computes the full Wigner D matrix for the Euler angle alpha, beta, and gamma with major index 'l' and minor index transition from m to mp. More...

Function Documentation

XLALSpinWeightedSphericalHarmonic()

COMPLEX16 XLALSpinWeightedSphericalHarmonic	(	REAL8	theta,
		REAL8	phi,
		int	s,
		int	l,
		int	m
	)

Computes the (s)Y(l,m) spin-weighted spherical harmonic.

From somewhere ....

See also: Implements Equations (II.9)-(II.13) of D. A. Brown, S. Fairhurst, B. Krishnan, R. A. Mercer, R. K. Kopparapu, L. Santamaria, and J. T. Whelan, "Data formats for numerical relativity waves", arXiv:0709.0093v1 (2007).

Currently only supports s=-2, l=2,3,4,5,6,7,8 modes.

Parameters

theta	polar angle (rad)
phi	azimuthal angle (rad)
s	spin weight
l	mode number l
m	mode number m

Definition at line 42 of file SphericalHarmonics.c.

XLALScalarSphericalHarmonic()

int XLALScalarSphericalHarmonic	(	COMPLEX16 *	y,
		UINT4	l,
		INT4	m,
		REAL8	theta,
		REAL8	phi
	)

Computes the scalar spherical harmonic \( Y_{lm}(\theta, \phi) \).

Parameters

y	output
l	value of l
m	value of m
theta	angle theta
phi	angle phi

Definition at line 430 of file SphericalHarmonics.c.

XLALSphHarm()

INT4 XLALSphHarm	(	COMPLEX16 *	out,
		UINT4	L,
		INT4	M,
		REAL4	theta,
		REAL4	phi
	)

Computes the spin 2 weighted spherical harmonic.

This function is now deprecated and will be removed soon. All calls should be replaced with calls to XLALSpinWeightedSphericalHarmonic().

Parameters

out	output
L	value of L
M	value of M
theta	angle with respect to the z axis
phi	angle with respect to the x axis

Definition at line 475 of file SphericalHarmonics.c.

XLALJacobiPolynomial()

double XLALJacobiPolynomial	(	int	n,
		int	alpha,
		int	beta,
		double	x
	)

Computes the n-th Jacobi polynomial for polynomial weights alpha and beta.

The implementation here is only valid for real x – enforced by the argument type. An extension to complex values would require evaluation of several gamma functions.

See http://en.wikipedia.org/wiki/Jacobi_polynomials

Definition at line 502 of file SphericalHarmonics.c.

XLALWignerdMatrix()

double XLALWignerdMatrix	(	int	l,
		int	mp,
		int	m,
		double	beta
	)

Parameters

l	mode number l
mp	mode number m'
m	mode number m
beta	euler angle (rad)

Definition at line 530 of file SphericalHarmonics.c.

XLALWignerDMatrix()

COMPLEX16 XLALWignerDMatrix	(	int	l,
		int	mp,
		int	m,
		double	alpha,
		double	beta,
		double	gam
	)

Computes the full Wigner D matrix for the Euler angle alpha, beta, and gamma with major index 'l' and minor index transition from m to mp.

Uses a slightly unconventional method since the intuitive version by Wigner is less suitable to algorthmic development.

See http://en.wikipedia.org/wiki/Wigner_D-matrix

Currently only supports the modes which are implemented for the spin weighted spherical harmonics.

Parameters

l	mode number l
mp	mode number m'
m	mode number m
alpha	euler angle (rad)
beta	euler angle (rad)
gam	euler angle (rad)

Definition at line 573 of file SphericalHarmonics.c.