111 lines
4.3 KiB
C
111 lines
4.3 KiB
C
/*! \file qpms_specfunc.h
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* \brief Various special and auxillary functions.
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*/
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#ifndef QPMS_SPECFUNC_H
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#define QPMS_SPECFUNC_H
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#include "qpms_types.h"
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#include <gsl/gsl_sf_legendre.h>
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/******************************************************************************
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* Spherical Bessel functions *
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******************************************************************************/
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// TODO unify types
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qpms_errno_t qpms_sph_bessel_fill(qpms_bessel_t typ, qpms_l_t lmax, complex double x, complex double *result_array);
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typedef struct {
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qpms_l_t lMax;
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double *akn; // coefficients as in DLMF 10.49.1
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//complex double *bkn; // coefficients of the derivatives
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} qpms_sbessel_calculator_t;
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qpms_sbessel_calculator_t *qpms_sbessel_calculator_init(void);
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void qpms_sbessel_calculator_pfree(qpms_sbessel_calculator_t *c);
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qpms_errno_t qpms_sbessel_calc_fill(qpms_sbessel_calculator_t *c, qpms_bessel_t typ, qpms_l_t lmax,
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double x, complex double *result_array);
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complex double qpms_sbessel_calc_h1(qpms_sbessel_calculator_t *c, qpms_l_t n, complex double x);
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qpms_errno_t qpms_sbessel_calc_h1_fill(qpms_sbessel_calculator_t *c, qpms_l_t lmax,
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complex double x, complex double *result_array);
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/******************************************************************************
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* Legendre functions and their "angular derivatives" *
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******************************************************************************/
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/*
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* N.B. for the norm definitions, see
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* https://www.gnu.org/software/gsl/manual/html_node/Associated-Legendre-Polynomials-and-Spherical-Harmonics.html
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* ( gsl/specfunc/legendre_source.c and 7.24.2 of gsl docs
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*/
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qpms_errno_t qpms_legendre_deriv_y_get(double **result, double **result_deriv, double x, qpms_l_t lMax,
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gsl_sf_legendre_t lnorm, double csphase); // free() result and result_deriv yourself!
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qpms_errno_t qpms_legendre_deriv_y_fill(double *where, double *where_deriv, double x,
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qpms_l_t lMax, gsl_sf_legendre_t lnorm, double csphase);
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double *qpms_legendre_y_get(double x, qpms_l_t lMax, qpms_normalisation_t norm);//NI
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double *qpms_legendre0d_y_get(qpms_l_t lMax, qpms_normalisation_t norm); //NI
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double *qpms_legendre_plus1d_y_get(qpms_l_t lMax, qpms_normalisation_t norm); //NI
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double *qpms_legendre_minus1d_y_get(qpms_l_t lMax, qpms_normalisation_t norm); //NI
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/// Array of Legendre and and auxillary \f$\pi_{lm}, \tau_{lm} \f$ functions.
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/**
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* See qpms_pitau_get() for definitions.
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*
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* The leg, pi, tau arrays are indexed using the standard qpms_mn2y() VSWF indexing.
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*/
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typedef struct {
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//qpms_normalisation_t norm;
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qpms_l_t lMax;
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//qpms_y_t nelem;
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double *leg, *pi, *tau;
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} qpms_pitau_t;
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/// Returns an array of normalised Legendre and auxillary \f$\pi_{lm}, \tau_{lm} \f$ functions.
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/**
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* The normalised Legendre function here is defined as
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* \f[
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* \Fer[norm.]{l}{m} = \csphase^{-1}
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* \sqrt{\frac{1}{l(l+1)}\frac{(l-m)!(2l+1)}{4\pi(l+m)!}},
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* \f] i.e. obtained using `gsl_sf_legendre_array_e()` with
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* `norm = GSL_SF_LEGENDRE_SPHARM` and divided by \f$ \sqrt{l(l+1)} \f$.
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*
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* The auxillary functions are defined as
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* \f[
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* \pi_{lm}(\cos \theta) = \frac{m}{\sin \theta} \Fer[norm.]{l}{m}(\cos\theta),\\
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* \tau_{lm}(\cos \theta) = \frac{\ud}{\ud \theta} \Fer[norm.]{l}{m}(\cos\theta)
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* \f]
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* with appropriate limit expression used if \f$ \abs{\cos\theta} = 1 \f$.
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*
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* When done, don't forget to deallocate the memory using qpms_pitau_free().
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*
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*/
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qpms_pitau_t qpms_pitau_get(double theta, qpms_l_t lMax, double csphase);
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/// Directly fills (pre-allocated) arrays of normalised Legendre and auxillary \f$\pi_{lm}, \tau_{lm} \f$ functions.
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/**
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* Arrays must be preallocated for `lMax * (lMax + 2)` elements. `NULL` targets are skipped.
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* For details, see qpms_pitau_get().
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*/
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qpms_errno_t qpms_pitau_fill(double *target_leg, double *target_pi, double *target_tau,
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double theta, qpms_l_t lMax, double csphase);
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/// Frees the dynamically allocated arrays from qpms_pitau_t.
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void qpms_pitau_free(qpms_pitau_t);
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//void qpms_pitau_pfree(qpms_pitau_t*);//NI
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// Associated Legendre polynomial at zero argument (DLMF 14.5.1) DEPRECATED?
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double qpms_legendre0(int m, int n);
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// Associated Legendre polynomial derivative at zero argument (DLMF 14.5.2)
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double qpms_legendred0(int m, int n);
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#endif // QPMS_SPECFUNC_H
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