qpms_specfunc.h File Reference

Various special and auxillary functions. More...

#include "qpms_types.h"
#include <gsl/gsl_sf_legendre.h>

Include dependency graph for qpms_specfunc.h:

Go to the source code of this file.

Data Structures
struct	qpms_sbessel_calculator_t
struct	qpms_pitau_t
	Array of Legendre and and auxillary \(\pi_{lm}, \tau_{lm} \) functions. More...

Functions
qpms_errno_t	qpms_sph_bessel_fill (qpms_bessel_t typ, qpms_l_t lmax, complex double x, complex double *result_array)
qpms_sbessel_calculator_t *	qpms_sbessel_calculator_init (void)
void	qpms_sbessel_calculator_pfree (qpms_sbessel_calculator_t *c)
qpms_errno_t	qpms_sbessel_calc_fill (qpms_sbessel_calculator_t *c, qpms_bessel_t typ, qpms_l_t lmax, double x, complex double *result_array)
complex double	qpms_sbessel_calc_h1 (qpms_sbessel_calculator_t *c, qpms_l_t n, complex double x)
qpms_errno_t	qpms_sbessel_calc_h1_fill (qpms_sbessel_calculator_t *c, qpms_l_t lmax, complex double x, complex double *result_array)
qpms_errno_t	qpms_legendre_deriv_y_get (double **result, double **result_deriv, double x, qpms_l_t lMax, gsl_sf_legendre_t lnorm, double csphase)
qpms_errno_t	qpms_legendre_deriv_y_fill (double *where, double *where_deriv, double x, qpms_l_t lMax, gsl_sf_legendre_t lnorm, double csphase)
double *	qpms_legendre_y_get (double x, qpms_l_t lMax, qpms_normalisation_t norm)
double *	qpms_legendre0d_y_get (qpms_l_t lMax, qpms_normalisation_t norm)
double *	qpms_legendre_plus1d_y_get (qpms_l_t lMax, qpms_normalisation_t norm)
double *	qpms_legendre_minus1d_y_get (qpms_l_t lMax, qpms_normalisation_t norm)
qpms_pitau_t	qpms_pitau_get (double theta, qpms_l_t lMax, double csphase)
	Returns an array of normalised Legendre and auxillary \(\pi_{lm}, \tau_{lm} \) functions. More...
qpms_errno_t	qpms_pitau_fill (double *target_leg, double *target_pi, double *target_tau, double theta, qpms_l_t lMax, double csphase)
	Directly fills (pre-allocated) arrays of normalised Legendre and auxillary \(\pi_{lm}, \tau_{lm} \) functions. More...
void	qpms_pitau_free (qpms_pitau_t)
	Frees the dynamically allocated arrays from qpms_pitau_t.
double	qpms_legendre0 (int m, int n)
double	qpms_legendred0 (int m, int n)

Detailed Description

Various special and auxillary functions.

Function Documentation

qpms_pitau_fill()

qpms_errno_t qpms_pitau_fill	(	double *	target_leg,
		double *	target_pi,
		double *	target_tau,
		double	theta,
		qpms_l_t	lMax,
		double	csphase
	)

Directly fills (pre-allocated) arrays of normalised Legendre and auxillary \(\pi_{lm}, \tau_{lm} \) functions.

Arrays must be preallocated for lMax * (lMax + 2) elements. NULL targets are skipped. For details, see qpms_pitau_get().

qpms_pitau_get()

qpms_pitau_t qpms_pitau_get	(	double	theta,
		qpms_l_t	lMax,
		double	csphase
	)

Returns an array of normalised Legendre and auxillary \(\pi_{lm}, \tau_{lm} \) functions.

The normalised Legendre function here is defined as

\[ \Fer[norm.]{l}{m} = \csphase^{-1} \sqrt{\frac{1}{l(l+1)}\frac{(l-m)!(2l+1)}{4\pi(l+m)!}}, \]

i.e. obtained using gsl_sf_legendre_array_e() with norm = GSL_SF_LEGENDRE_SPHARM and divided by \( \sqrt{l(l+1)} \).

The auxillary functions are defined as

\[ \pi_{lm}(\cos \theta) = \frac{m}{\sin \theta} \Fer[norm.]{l}{m}(\cos\theta),\\ \tau_{lm}(\cos \theta) = \frac{\ud}{\ud \theta} \Fer[norm.]{l}{m}(\cos\theta) \]

with appropriate limit expression used if \( \abs{\cos\theta} = 1 \).

When done, don't forget to deallocate the memory using qpms_pitau_free().