qpms/qpms_specfunc.h

   1 /*! \file qpms_specfunc.h
   2  * \brief Various special and auxillary functions.
   3  */
   4 #ifndef QPMS_SPECFUNC_H
   5 #define QPMS_SPECFUNC_H
   6 #include "qpms_types.h"
   7 #include <gsl/gsl_sf_legendre.h>
   8
   9 /******************************************************************************
  10  *                    Spherical Bessel functions                              *
  11  ******************************************************************************/
  12
  13 // TODO unify types
  14 qpms_errno_t qpms_sph_bessel_fill(qpms_bessel_t typ, qpms_l_t lmax, complex double x, complex double *result_array);
  15
  16
  17
  18 typedef struct {
  19         qpms_l_t lMax;
  20         double *akn; // coefficients as in DLMF 10.49.1
  21         //complex double *bkn; // coefficients of the derivatives
  22 } qpms_sbessel_calculator_t;
  23
  24 qpms_sbessel_calculator_t *qpms_sbessel_calculator_init(void);
  25 void qpms_sbessel_calculator_pfree(qpms_sbessel_calculator_t *c);
  26
  27 qpms_errno_t qpms_sbessel_calc_fill(qpms_sbessel_calculator_t *c, qpms_bessel_t typ, qpms_l_t lmax,
  28         double x, complex double *result_array);
  29
  30 complex double qpms_sbessel_calc_h1(qpms_sbessel_calculator_t *c, qpms_l_t n, complex double x);
  31 qpms_errno_t qpms_sbessel_calc_h1_fill(qpms_sbessel_calculator_t *c, qpms_l_t lmax,
  32                 complex double x, complex double *result_array);
  33
  34
  35 /******************************************************************************
  36  *         Legendre functions and their "angular derivatives"                 *
  37  ******************************************************************************/
  38
  39 /*
  40  * N.B. for the norm definitions, see
  41  * https://www.gnu.org/software/gsl/manual/html_node/Associated-Legendre-Polynomials-and-Spherical-Harmonics.html
  42  * ( gsl/specfunc/legendre_source.c and 7.24.2 of gsl docs
  43  */
  44
  45 qpms_errno_t qpms_legendre_deriv_y_get(double **result, double **result_deriv, double x, qpms_l_t lMax,
  46                 gsl_sf_legendre_t lnorm, double csphase); // free() result and result_deriv yourself!
  47 qpms_errno_t qpms_legendre_deriv_y_fill(double *where, double *where_deriv, double x,
  48                 qpms_l_t lMax, gsl_sf_legendre_t lnorm, double csphase);
  49
  50
  51 double *qpms_legendre_y_get(double x, qpms_l_t lMax, qpms_normalisation_t norm);//NI
  52 double *qpms_legendre0d_y_get(qpms_l_t lMax, qpms_normalisation_t norm); //NI
  53 double *qpms_legendre_plus1d_y_get(qpms_l_t lMax, qpms_normalisation_t norm); //NI
  54 double *qpms_legendre_minus1d_y_get(qpms_l_t lMax, qpms_normalisation_t norm); //NI
  55
  56
  57
  58 /// Array of Legendre and and auxillary \f$\pi_{lm}, \tau_{lm} \f$ functions.
  59 /**
  60  * See qpms_pitau_get() for definitions.
  61  *
  62  * The leg, pi, tau arrays are indexed using the standard qpms_mn2y() VSWF indexing.
  63  */
  64 typedef struct {
  65         //qpms_normalisation_t norm;
  66         qpms_l_t lMax;
  67         //qpms_y_t nelem;
  68         double *leg, *pi, *tau;
  69 } qpms_pitau_t;
  70
  71 /// Returns an array of normalised Legendre and auxillary \f$\pi_{lm}, \tau_{lm} \f$ functions.
  72 /**
  73  * The normalised Legendre function here is defined as
  74  * \f[
  75  *      \Fer[norm.]{l}{m} = \csphase^{-1}
  76  *              \sqrt{\frac{1}{l(l+1)}\frac{(l-m)!(2l+1)}{4\pi(l+m)!}},
  77  * \f] i.e. obtained using `gsl_sf_legendre_array_e()` with
  78  * `norm = GSL_SF_LEGENDRE_SPHARM` and divided by \f$ \sqrt{l(l+1)} \f$.
  79  *
  80  * The auxillary functions are defined as
  81  * \f[
  82  *      \pi_{lm}(\cos \theta) = \frac{m}{\sin \theta} \Fer[norm.]{l}{m}(\cos\theta),\\
  83  *      \tau_{lm}(\cos \theta) = \frac{\ud}{\ud \theta} \Fer[norm.]{l}{m}(\cos\theta)
  84  * \f]
  85  * with appropriate limit expression used if \f$ \abs{\cos\theta} = 1 \f$.
  86  *
  87  * When done, don't forget to deallocate the memory using qpms_pitau_free().
  88  *
  89  */
  90 qpms_pitau_t qpms_pitau_get(double theta, qpms_l_t lMax, double csphase);
  91
  92 /// Directly fills (pre-allocated) arrays of normalised Legendre and auxillary \f$\pi_{lm}, \tau_{lm} \f$ functions.
  93 /**
  94  * Arrays must be preallocated for `lMax * (lMax + 2)` elements. `NULL` targets are skipped.
  95  * For details, see qpms_pitau_get().
  96  */
  97 qpms_errno_t qpms_pitau_fill(double *target_leg, double *target_pi, double *target_tau,
  98                 double theta, qpms_l_t lMax, double csphase);
  99
 100 /// Frees the dynamically allocated arrays from qpms_pitau_t.
 101 void qpms_pitau_free(qpms_pitau_t);
 102
 103 //void qpms_pitau_pfree(qpms_pitau_t*);//NI
 104
 105 // Associated Legendre polynomial at zero argument (DLMF 14.5.1) DEPRECATED?
 106 double qpms_legendre0(int m, int n);
 107 // Associated Legendre polynomial derivative at zero argument (DLMF 14.5.2)
 108 double qpms_legendred0(int m, int n);
 109
 110 #endif // QPMS_SPECFUNC_H