/* specfunc/legendre_con.c

 * 

 * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman

 * Copyright (C) 2010 Brian Gough

 * 

 * This program is free software; you can redistribute it and/or modify

 * it under the terms of the GNU General Public License as published by

 * the Free Software Foundation; either version 3 of the License, or (at

 * your option) any later version.

 * 

 * This program is distributed in the hope that it will be useful, but

 * WITHOUT ANY WARRANTY; without even the implied warranty of

 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU

 * General Public License for more details.

 * 

 * You should have received a copy of the GNU General Public License

 * along with this program; if not, write to the Free Software

 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.

 */

/* Author:  G. Jungman */

#include <config.h>

#include <gsl/gsl_math.h>

#include <gsl/gsl_errno.h>

#include <gsl/gsl_poly.h>

#include <gsl/gsl_sf_exp.h>

#include <gsl/gsl_sf_trig.h>

#include <gsl/gsl_sf_gamma.h>

#include <gsl/gsl_sf_ellint.h>

#include <gsl/gsl_sf_pow_int.h>

#include <gsl/gsl_sf_bessel.h>

#include <gsl/gsl_sf_hyperg.h>

#include <gsl/gsl_sf_legendre.h>

#include "error.h"

#include "legendre.h"

#define Root_2OverPi_  0.797884560802865355879892

#define locEPS         (1000.0*GSL_DBL_EPSILON)

/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/

#define RECURSE_LARGE  (1.0e-5*GSL_DBL_MAX)

#define RECURSE_SMALL  (1.0e+5*GSL_DBL_MIN)

/* Continued fraction for f_{ell+1}/f_ell

 * f_ell := P^{-mu-ell}_{-1/2 + I tau}(x),  x < 1.0

 *

 * Uses standard CF method from Temme's book.

 */

static

int

conicalP_negmu_xlt1_CF1(const double mu, const int ell, const double tau,

                        const double x, gsl_sf_result * result)

{

  const double RECUR_BIG = GSL_SQRT_DBL_MAX;

  const int maxiter = 5000;

  int n = 1;

  double xi = x/(sqrt(1.0-x)*sqrt(1.0+x));

  double Anm2 = 1.0;

  double Bnm2 = 0.0;

  double Anm1 = 0.0;

  double Bnm1 = 1.0;

  double a1 = 1.0;

  double b1 = 2.0*(mu + ell + 1.0) * xi;

  double An = b1*Anm1 + a1*Anm2;

  double Bn = b1*Bnm1 + a1*Bnm2;

  double an, bn;

  double fn = An/Bn;

  while(n < maxiter) {

    double old_fn;

    double del;

    n++;

    Anm2 = Anm1;

    Bnm2 = Bnm1;

    Anm1 = An;

    Bnm1 = Bn;

    an = tau*tau + (mu - 0.5 + ell + n)*(mu - 0.5 + ell + n);

    bn = 2.0*(ell + mu + n) * xi;

    An = bn*Anm1 + an*Anm2;

    Bn = bn*Bnm1 + an*Bnm2;

    if(fabs(An) > RECUR_BIG || fabs(Bn) > RECUR_BIG) {

      An /= RECUR_BIG;

      Bn /= RECUR_BIG;

      Anm1 /= RECUR_BIG;

      Bnm1 /= RECUR_BIG;

      Anm2 /= RECUR_BIG;

      Bnm2 /= RECUR_BIG;

    }

    old_fn = fn;

    fn = An/Bn;

    del = old_fn/fn;

    if(fabs(del - 1.0) < 2.0*GSL_DBL_EPSILON) break;

  }

  result->val = fn;

  result->err = 4.0 * GSL_DBL_EPSILON * (sqrt(n) + 1.0) * fabs(fn);

  if(n >= maxiter)

    GSL_ERROR ("error", GSL_EMAXITER);

  else

    return GSL_SUCCESS;

}

/* Continued fraction for f_{ell+1}/f_ell

 * f_ell := P^{-mu-ell}_{-1/2 + I tau}(x),  x >= 1.0

 *

 * Uses Gautschi (Euler) equivalent series.

 */

static

int

conicalP_negmu_xgt1_CF1(const double mu, const int ell, const double tau,

                        const double x, gsl_sf_result * result)

{ 

  const int maxk = 20000;

  const double gamma = 1.0-1.0/(x*x);

  const double pre = sqrt(x-1.0)*sqrt(x+1.0) / (x*(2.0*(ell+mu+1.0)));

  double tk   = 1.0;

  double sum  = 1.0;

  double rhok = 0.0;

  int k;

  for(k=1; k

    double tlk = 2.0*(ell + mu + k);

    double l1k = (ell + mu - 0.5 + 1.0 + k);

    double ak = -(tau*tau + l1k*l1k)/(tlk*(tlk+2.0)) * gamma;

    rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok));

    tk  *= rhok;

    sum += tk;

    if(fabs(tk/sum) < GSL_DBL_EPSILON) break;

  }

  result->val  = pre * sum;

  result->err  = fabs(pre * tk);

  result->err += 2.0 * GSL_DBL_EPSILON * (sqrt(k) + 1.0) * fabs(pre*sum);

  if(k >= maxk)

    GSL_ERROR ("error", GSL_EMAXITER);

  else

    return GSL_SUCCESS;

}

/* Implementation of large negative mu asymptotic

 * [Dunster, Proc. Roy. Soc. Edinburgh 119A, 311 (1991), p. 326]

 */

inline

static double olver_U1(double beta2, double p)

{

  return (p-1.0)/(24.0*(1.0+beta2)) * (3.0 + beta2*(2.0 + 5.0*p*(1.0+p)));

}

inline

static double olver_U2(double beta2, double p)

{

  double beta4 = beta2*beta2;

  double p2    = p*p;

  double poly1 =  4.0*beta4 + 84.0*beta2 - 63.0;

  double poly2 = 16.0*beta4 + 90.0*beta2 - 81.0;

  double poly3 = beta2*p2*(97.0*beta2 - 432.0 + 77.0*p*(beta2-6.0) - 385.0*beta2*p2*(1.0 + p));

  return (1.0-p)/(1152.0*(1.0+beta2)) * (poly1 + poly2 + poly3);

}

static const double U3c1[] = {   -1307.0,   -1647.0,    3375.0,    3675.0 };

static const double U3c2[] = {   29366.0,   35835.0, -252360.0, -272630.0,

                                276810.0,  290499.0 };

static const double U3c3[] = {  -29748.0,   -8840.0, 1725295.0, 1767025.0,

                              -7313470.0, -754778.0, 6309875.0, 6480045.0 };

static const double U3c4[] = {    2696.0,    -16740.0,   -524250.0,  -183975.0,

                              14670540.0,  14172939.0, -48206730.0, -48461985.0,

                              36756720.0,  37182145.0 };

static const double U3c5[] = {       9136.0,      22480.0,     12760.0,

                                  -252480.0,   -4662165.0,   -1705341.0,

                                 92370135.0,   86244015.0, -263678415.0,

                               -260275015.0, 185910725.0,  185910725.0 };

#if 0

static double olver_U3(double beta2, double p)

{

  double beta4 = beta2*beta2;

  double beta6 = beta4*beta2;

  double opb2s = (1.0+beta2)*(1.0+beta2);

  double den   = 39813120.0 * opb2s*opb2s;

  double poly1 = gsl_poly_eval(U3c1, 4, p);

  double poly2 = gsl_poly_eval(U3c2, 6, p);

  double poly3 = gsl_poly_eval(U3c3, 8, p);

  double poly4 = gsl_poly_eval(U3c4, 10, p);

  double poly5 = gsl_poly_eval(U3c5, 12, p);

  return (p-1.0)*(     1215.0*poly1 + 324.0*beta2*poly2

                 + 54.0*beta4*poly3 +  12.0*beta6*poly4

                 + beta4*beta4*poly5

                 ) / den;

}

#endif /* 0 */

/* Large negative mu asymptotic

 * P^{-mu}_{-1/2 + I tau}, mu -> Inf

 * |x| < 1

 *

 * [Dunster, Proc. Roy. Soc. Edinburgh 119A, 311 (1991), p. 326]

 */

int

gsl_sf_conicalP_xlt1_large_neg_mu_e(double mu, double tau, double x,

                                       gsl_sf_result * result, double * ln_multiplier)

{

  double beta  = tau/mu;

  double beta2 = beta*beta;

  double S     = beta * acos((1.0-beta2)/(1.0+beta2));

  double p     = x/sqrt(beta2*(1.0-x*x) + 1.0);

  gsl_sf_result lg_mup1;

  int lg_stat = gsl_sf_lngamma_e(mu+1.0, &lg_mup1);

  double ln_pre_1 =  0.5*mu*(S - log(1.0+beta2) + log((1.0-p)/(1.0+p))) - lg_mup1.val;

  double ln_pre_2 = -0.25 * log(1.0 + beta2*(1.0-x));

  double ln_pre_3 = -tau * atan(p*beta);

  double ln_pre = ln_pre_1 + ln_pre_2 + ln_pre_3;

  double sum   = 1.0 - olver_U1(beta2, p)/mu + olver_U2(beta2, p)/(mu*mu);

  if(sum == 0.0) {

    result->val = 0.0;

    result->err = 0.0;

    *ln_multiplier = 0.0;

    return GSL_SUCCESS;

  }

  else {

    int stat_e = gsl_sf_exp_mult_e(ln_pre, sum, result);

    if(stat_e != GSL_SUCCESS) {

      result->val = sum;

      result->err = 2.0 * GSL_DBL_EPSILON * fabs(sum);

      *ln_multiplier = ln_pre;

    }

    else {

      *ln_multiplier = 0.0;

    }

    return lg_stat;

  }

}

/* Implementation of large tau asymptotic

 *

 * A_n^{-mu}, B_n^{-mu}  [Olver, p.465, 469]

 */

inline

static double olver_B0_xi(double mu, double xi)

{

  return (1.0 - 4.0*mu*mu)/(8.0*xi) * (1.0/tanh(xi) - 1.0/xi);

}

static double olver_A1_xi(double mu, double xi, double x)

{

  double B = olver_B0_xi(mu, xi);

  double psi;

  if(fabs(x - 1.0) < GSL_ROOT4_DBL_EPSILON) {

    double y = x - 1.0;

    double s = -1.0/3.0 + y*(2.0/15.0 - y *(61.0/945.0 - 452.0/14175.0*y));

    psi = (4.0*mu*mu - 1.0)/16.0 * s;

  }

  else {

    psi = (4.0*mu*mu - 1.0)/16.0 * (1.0/(x*x-1.0) - 1.0/(xi*xi));

  }

  return 0.5*xi*xi*B*B + (mu+0.5)*B - psi + mu/6.0*(0.25 - mu*mu);

}

inline

static double olver_B0_th(double mu, double theta)

{

  return -(1.0 - 4.0*mu*mu)/(8.0*theta) * (1.0/tan(theta) - 1.0/theta);

}

static double olver_A1_th(double mu, double theta, double x)

{

  double B = olver_B0_th(mu, theta);

  double psi;

  if(fabs(x - 1.0) < GSL_ROOT4_DBL_EPSILON) {

    double y = 1.0 - x;

    double s = -1.0/3.0 + y*(2.0/15.0 - y *(61.0/945.0 - 452.0/14175.0*y));

    psi = (4.0*mu*mu - 1.0)/16.0 * s;

  }

  else {

    psi = (4.0*mu*mu - 1.0)/16.0 * (1.0/(x*x-1.0) + 1.0/(theta*theta));

  }

  return -0.5*theta*theta*B*B + (mu+0.5)*B - psi + mu/6.0*(0.25 - mu*mu);

}

/* Large tau uniform asymptotics

 * P^{-mu}_{-1/2 + I tau}

 * 1 < x

 * tau -> Inf 

 * [Olver, p. 469]

 */

int

gsl_sf_conicalP_xgt1_neg_mu_largetau_e(const double mu, const double tau,

                                          const double x, double acosh_x,

                                          gsl_sf_result * result, double * ln_multiplier)

{

  double xi = acosh_x;

  double ln_xi_pre;

  double ln_pre;

  double sumA, sumB, sum;

  double arg;

  gsl_sf_result J_mup1;

  gsl_sf_result J_mu;

  double J_mum1;

  if(xi < GSL_ROOT4_DBL_EPSILON) {

    ln_xi_pre = -xi*xi/6.0;           /* log(1.0 - xi*xi/6.0) */

  }

  else {

    gsl_sf_result lnshxi;

    gsl_sf_lnsinh_e(xi, &lnshxi);

    ln_xi_pre = log(xi) - lnshxi.val;     /* log(xi/sinh(xi) */

  }

  ln_pre = 0.5*ln_xi_pre - mu*log(tau);

  arg = tau*xi;

  gsl_sf_bessel_Jnu_e(mu + 1.0,   arg, &J_mup1);

  gsl_sf_bessel_Jnu_e(mu,         arg, &J_mu);

  J_mum1 = -J_mup1.val + 2.0*mu/arg*J_mu.val;      /* careful of mu < 1 */

  sumA = 1.0 - olver_A1_xi(-mu, xi, x)/(tau*tau);

  sumB = olver_B0_xi(-mu, xi);

  sum  = J_mu.val * sumA - xi/tau * J_mum1 * sumB;

  if(sum == 0.0) {

    result->val = 0.0;

    result->err = 0.0;

    *ln_multiplier = 0.0;

    return GSL_SUCCESS;

  }

  else {

    int stat_e = gsl_sf_exp_mult_e(ln_pre, sum, result);

    if(stat_e != GSL_SUCCESS) {

      result->val = sum;

      result->err = 2.0 * GSL_DBL_EPSILON * fabs(sum);

      *ln_multiplier = ln_pre;

    }

    else {

      *ln_multiplier = 0.0;

    }

    return GSL_SUCCESS;

  }

}

/* Large tau uniform asymptotics

 * P^{-mu}_{-1/2 + I tau}

 * -1 < x < 1

 * tau -> Inf 

 * [Olver, p. 473]

 */

int

gsl_sf_conicalP_xlt1_neg_mu_largetau_e(const double mu, const double tau,

                                          const double x, const double acos_x,

                                          gsl_sf_result * result, double * ln_multiplier)

{

  double theta = acos_x;

  double ln_th_pre;

  double ln_pre;

  double sumA, sumB, sum, sumerr;

  double arg;

  gsl_sf_result I_mup1, I_mu;

  double I_mum1;

  if(theta < GSL_ROOT4_DBL_EPSILON) {

    ln_th_pre = theta*theta/6.0;   /* log(1.0 + theta*theta/6.0) */

  }

  else {

    ln_th_pre = log(theta/sin(theta));

  }

  ln_pre = 0.5 * ln_th_pre - mu * log(tau);

  arg = tau*theta;

  gsl_sf_bessel_Inu_e(mu + 1.0,   arg, &I_mup1);

  gsl_sf_bessel_Inu_e(mu,         arg, &I_mu);

  I_mum1 = I_mup1.val + 2.0*mu/arg * I_mu.val; /* careful of mu < 1 */

  sumA = 1.0 - olver_A1_th(-mu, theta, x)/(tau*tau);

  sumB = olver_B0_th(-mu, theta);

  sum  = I_mu.val * sumA - theta/tau * I_mum1 * sumB;

  sumerr  = fabs(I_mu.err * sumA);

  sumerr += fabs(I_mup1.err * theta/tau * sumB);

  sumerr += fabs(I_mu.err   * theta/tau * sumB * 2.0 * mu/arg);

  if(sum == 0.0) {

    result->val = 0.0;

    result->err = 0.0;

    *ln_multiplier = 0.0;

    return GSL_SUCCESS;

  }

  else {

    int stat_e = gsl_sf_exp_mult_e(ln_pre, sum, result);

    if(stat_e != GSL_SUCCESS) {

      result->val  = sum;

      result->err  = sumerr;

      result->err += GSL_DBL_EPSILON * fabs(sum);

      *ln_multiplier = ln_pre;

    }

    else {

      *ln_multiplier = 0.0;

    }

    return GSL_SUCCESS;

  }

}

/* Hypergeometric function which appears in the

 * large x expansion below:

 *

 *   2F1(1/4 - mu/2 - I tau/2, 3/4 - mu/2 - I tau/2, 1 - I tau, y)

 *

 * Note that for the usage below y = 1/x^2;

 */

static

int

conicalP_hyperg_large_x(const double mu, const double tau, const double y,

                        double * reF, double * imF)

{

  const int kmax = 1000;

  const double re_a = 0.25 - 0.5*mu;

  const double re_b = 0.75 - 0.5*mu;

  const double re_c = 1.0;

  const double im_a = -0.5*tau;

  const double im_b = -0.5*tau;

  const double im_c = -tau;

  double re_sum = 1.0;

  double im_sum = 0.0;

  double re_term = 1.0;

  double im_term = 0.0;

  int k;

  for(k=1; k<=kmax; k++) {

    double re_ak = re_a + k - 1.0;

    double re_bk = re_b + k - 1.0;

    double re_ck = re_c + k - 1.0;

    double im_ak = im_a;

    double im_bk = im_b;

    double im_ck = im_c;

    double den = re_ck*re_ck + im_ck*im_ck;

    double re_multiplier = ((re_ak*re_bk - im_ak*im_bk)*re_ck + im_ck*(im_ak*re_bk + re_ak*im_bk)) / den;

    double im_multiplier = ((im_ak*re_bk + re_ak*im_bk)*re_ck - im_ck*(re_ak*re_bk - im_ak*im_bk)) / den;

    double re_tmp = re_multiplier*re_term - im_multiplier*im_term;

    double im_tmp = im_multiplier*re_term + re_multiplier*im_term;

    double asum = fabs(re_sum) + fabs(im_sum);

    re_term = y/k * re_tmp;

    im_term = y/k * im_tmp;

    if(fabs(re_term/asum) < GSL_DBL_EPSILON && fabs(im_term/asum) < GSL_DBL_EPSILON) break;

    re_sum += re_term;

    im_sum += im_term;

  }

  *reF = re_sum;

  *imF = im_sum;

  if(k == kmax)

    GSL_ERROR ("error", GSL_EMAXITER);

  else  

    return GSL_SUCCESS;

}

/* P^{mu}_{-1/2 + I tau}

 * x->Inf

 */

int

gsl_sf_conicalP_large_x_e(const double mu, const double tau, const double x,

                             gsl_sf_result * result, double * ln_multiplier)

{

  /* 2F1 term

   */

  double y = ( x < 0.5*GSL_SQRT_DBL_MAX ? 1.0/(x*x) : 0.0 );

  double reF, imF;

  int stat_F = conicalP_hyperg_large_x(mu, tau, y, &reF, &imF;;

  /* f = Gamma(+i tau)/Gamma(1/2 - mu + i tau)

   * FIXME: shift so it's better for tau-> 0

   */

  gsl_sf_result lgr_num, lgth_num;

  gsl_sf_result lgr_den, lgth_den;

  int stat_gn = gsl_sf_lngamma_complex_e(0.0,tau,&lgr_num,&lgth_num);

  int stat_gd = gsl_sf_lngamma_complex_e(0.5-mu,tau,&lgr_den,&lgth_den);

  double angle = lgth_num.val - lgth_den.val + atan2(imF,reF);

  double lnx   = log(x);

  double lnxp1 = log(x+1.0);

  double lnxm1 = log(x-1.0);

  double lnpre_const = 0.5*M_LN2 - 0.5*M_LNPI;

  double lnpre_comm = (mu-0.5)*lnx - 0.5*mu*(lnxp1 + lnxm1);

  double lnpre_err  =   GSL_DBL_EPSILON * (0.5*M_LN2 + 0.5*M_LNPI)

                      + GSL_DBL_EPSILON * fabs((mu-0.5)*lnx)

                      + GSL_DBL_EPSILON * fabs(0.5*mu)*(fabs(lnxp1)+fabs(lnxm1));

  /*  result = pre*|F|*|f| * cos(angle - tau * (log(x)+M_LN2))

   */

  gsl_sf_result cos_result;

  int stat_cos = gsl_sf_cos_e(angle + tau*(log(x) + M_LN2), &cos_result);

  int status = GSL_ERROR_SELECT_4(stat_cos, stat_gd, stat_gn, stat_F);

  if(cos_result.val == 0.0) {

    result->val = 0.0;

    result->err = 0.0;

    return status;

  }

  else {

    double lnFf_val = 0.5*log(reF*reF+imF*imF) + lgr_num.val - lgr_den.val;

    double lnFf_err = lgr_num.err + lgr_den.err + GSL_DBL_EPSILON * fabs(lnFf_val);

    double lnnoc_val = lnpre_const + lnpre_comm + lnFf_val;

    double lnnoc_err = lnpre_err + lnFf_err + GSL_DBL_EPSILON * fabs(lnnoc_val);

    int stat_e = gsl_sf_exp_mult_err_e(lnnoc_val, lnnoc_err,

                                          cos_result.val, cos_result.err,

                                          result);

    if(stat_e == GSL_SUCCESS) {

      *ln_multiplier = 0.0;

    }

    else {

      result->val  = cos_result.val;

      result->err  = cos_result.err;

      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

      *ln_multiplier = lnnoc_val;

    }

    return status;

  }

}

/* P^{mu}_{-1/2 + I tau}  first hypergeometric representation

 * -1 < x < 1

 * This is more effective for |x| small, however it will work w/o

 * reservation for any x < 0 because everything is positive

 * definite in that case.

 *

 * [Kolbig,   (3)] (note typo in args of gamma functions)

 * [Bateman, (22)] (correct form)

 */

static

int

conicalP_xlt1_hyperg_A(double mu, double tau, double x, gsl_sf_result * result)

{

  double x2 = x*x;

  double err_amp = 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-fabs(x)));

  double pre_val = M_SQRTPI / pow(0.5*sqrt(1-x2), mu);

  double pre_err = err_amp * GSL_DBL_EPSILON * (fabs(mu)+1.0) * fabs(pre_val) ;

  gsl_sf_result ln_g1, ln_g2, arg_g1, arg_g2;

  gsl_sf_result F1, F2;

  gsl_sf_result pre1, pre2;

  double t1_val, t1_err;

  double t2_val, t2_err;

  int stat_F1 = gsl_sf_hyperg_2F1_conj_e(0.25 - 0.5*mu, 0.5*tau, 0.5, x2, &F1;;

  int stat_F2 = gsl_sf_hyperg_2F1_conj_e(0.75 - 0.5*mu, 0.5*tau, 1.5, x2, &F2;;

  int status = GSL_ERROR_SELECT_2(stat_F1, stat_F2);

  gsl_sf_lngamma_complex_e(0.75 - 0.5*mu, -0.5*tau, &ln_g1, &arg_g1);

  gsl_sf_lngamma_complex_e(0.25 - 0.5*mu, -0.5*tau, &ln_g2, &arg_g2);

  gsl_sf_exp_err_e(-2.0*ln_g1.val, 2.0*ln_g1.err, &pre1;;

  gsl_sf_exp_err_e(-2.0*ln_g2.val, 2.0*ln_g2.err, &pre2;;

  pre2.val *= -2.0*x;

  pre2.err *=  2.0*fabs(x);

  pre2.err +=  GSL_DBL_EPSILON * fabs(pre2.val);

  t1_val = pre1.val * F1.val;

  t1_err = fabs(pre1.val) * F1.err + pre1.err * fabs(F1.val);

  t2_val = pre2.val * F2.val;

  t2_err = fabs(pre2.val) * F2.err + pre2.err * fabs(F2.val);

  result->val  = pre_val * (t1_val + t2_val);

  result->err  = pre_val * (t1_err + t2_err);

  result->err += pre_err * fabs(t1_val + t2_val);

  result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

  return status;

}

/* P^{mu}_{-1/2 + I tau}

 * defining hypergeometric representation

 * [Abramowitz+Stegun, 8.1.2]

 * 1 < x < 3

 * effective for x near 1

 *

 */

#if 0

static

int

conicalP_def_hyperg(double mu, double tau, double x, double * result)

{

  double F;

  int stat_F = gsl_sf_hyperg_2F1_conj_renorm_e(0.5, tau, 1.0-mu, 0.5*(1.0-x), &F);

  *result = pow((x+1.0)/(x-1.0), 0.5*mu) * F;

  return stat_F;

}

#endif /* 0 */

/* P^{mu}_{-1/2 + I tau}  second hypergeometric representation

 * [Zhurina+Karmazina, (3.1)] 

 * -1 < x < 3

 * effective for x near 1

 *

 */

#if 0

static

int

conicalP_xnear1_hyperg_C(double mu, double tau, double x, double * result)

{

  double ln_pre, arg_pre;

  double ln_g1, arg_g1;

  double ln_g2, arg_g2;

  double F;

  int stat_F = gsl_sf_hyperg_2F1_conj_renorm_e(0.5+mu, tau, 1.0+mu, 0.5*(1.0-x), &F);

  gsl_sf_lngamma_complex_e(0.5+mu, tau, &ln_g1, &arg_g1);

  gsl_sf_lngamma_complex_e(0.5-mu, tau, &ln_g2, &arg_g2);

  ln_pre  = mu*M_LN2 - 0.5*mu*log(fabs(x*x-1.0)) + ln_g1 - ln_g2;

  arg_pre = arg_g1 - arg_g2;

  *result = exp(ln_pre) * F;

  return stat_F;

}

#endif /* 0 */

/* V0, V1 from Kolbig, m = 0

 */

static

int

conicalP_0_V(const double t, const double f, const double tau, const double sgn,

             double * V0, double * V1)

{

  double C[8];

  double T[8];

  double H[8];

  double V[12];

  int i;

  T[0] = 1.0;

  H[0] = 1.0;

  V[0] = 1.0;

  for(i=1; i<=7; i++) {

    T[i] = T[i-1] * t;

    H[i] = H[i-1] * (t*f);

  }

  for(i=1; i<=11; i++) {

    V[i] = V[i-1] * tau;

  }

  C[0] = 1.0;

  C[1] = (H[1]-1.0)/(8.0*T[1]);

  C[2] = (9.0*H[2] + 6.0*H[1] - 15.0 - sgn*8.0*T[2])/(128.0*T[2]);

  C[3] = 5.0*(15.0*H[3] + 27.0*H[2] + 21.0*H[1] - 63.0 - sgn*T[2]*(16.0*H[1]+24.0))/(1024.0*T[3]);

  C[4] = 7.0*(525.0*H[4] + 1500.0*H[3] + 2430.0*H[2] + 1980.0*H[1] - 6435.0

              + 192.0*T[4] - sgn*T[2]*(720.0*H[2]+1600.0*H[1]+2160.0)

              ) / (32768.0*T[4]);

  C[5] = 21.0*(2835.0*H[5] + 11025.0*H[4] + 24750.0*H[3] + 38610.0*H[2]

               + 32175.0*H[1] - 109395.0 + T[4]*(1984.0*H[1]+4032.0)

               - sgn*T[2]*(4800.0*H[3]+15120.0*H[2]+26400.0*H[1]+34320.0)

               ) / (262144.0*T[5]);

  C[6] = 11.0*(218295.0*H[6] + 1071630.0*H[5] + 3009825.0*H[4] + 6142500.0*H[3]

               + 9398025.0*H[2] + 7936110.0*H[1] - 27776385.0

               + T[4]*(254016.0*H[2]+749952.0*H[1]+1100736.0)

               - sgn*T[2]*(441000.0*H[4] + 1814400.0*H[3] + 4127760.0*H[2]

                         + 6552000.0*H[1] + 8353800.0 + 31232.0*T[4]

                         )

               ) / (4194304.0*T[6]);

  *V0 = C[0] + (-4.0*C[3]/T[1]+C[4])/V[4]

             + (-192.0*C[5]/T[3]+144.0*C[6]/T[2])/V[8]

             + sgn * (-C[2]/V[2]

                      + (-24.0*C[4]/T[2]+12.0*C[5]/T[1]-C[6])/V[6] 

                      + (-1920.0*C[6]/T[4])/V[10]

                      );

  *V1 = C[1]/V[1] + (8.0*(C[3]/T[2]-C[4]/T[1])+C[5])/V[5]

                  + (384.0*C[5]/T[4] - 768.0*C[6]/T[3])/V[9]

                  + sgn * ((2.0*C[2]/T[1]-C[3])/V[3]

                           + (48.0*C[4]/T[3]-72.0*C[5]/T[2] + 18.0*C[6]/T[1])/V[7]

                           + (3840.0*C[6]/T[5])/V[11]

                           );

  return GSL_SUCCESS;

}

/* V0, V1 from Kolbig, m = 1

 */

static

int

conicalP_1_V(const double t, const double f, const double tau, const double sgn,

             double * V0, double * V1)

{

  double Cm1;

  double C[8];

  double T[8];

  double H[8];

  double V[12];

  int i;

  T[0] = 1.0;

  H[0] = 1.0;

  V[0] = 1.0;

  for(i=1; i<=7; i++) {

    T[i] = T[i-1] * t;

    H[i] = H[i-1] * (t*f);

  }

  for(i=1; i<=11; i++) {

    V[i] = V[i-1] * tau;

  }

  Cm1  = -1.0;

  C[0] = 3.0*(1.0-H[1])/(8.0*T[1]);

  C[1] = (-15.0*H[2]+6.0*H[1]+9.0+sgn*8.0*T[2])/(128.0*T[2]);

  C[2] = 3.0*(-35.0*H[3] - 15.0*H[2] + 15.0*H[1] + 35.0 + sgn*T[2]*(32.0*H[1]+8.0))/(1024.0*T[3]);

  C[3] = (-4725.0*H[4] - 6300.0*H[3] - 3150.0*H[2] + 3780.0*H[1] + 10395.0

          -1216.0*T[4] + sgn*T[2]*(6000.0*H[2]+5760.0*H[1]+1680.0)) / (32768.0*T[4]);

  C[4] = 7.0*(-10395.0*H[5] - 23625.0*H[4] - 28350.0*H[3] - 14850.0*H[2]

              +19305.0*H[1] + 57915.0 - T[4]*(6336.0*H[1]+6080.0)

              + sgn*T[2]*(16800.0*H[3] + 30000.0*H[2] + 25920.0*H[1] + 7920.0)

              ) / (262144.0*T[5]);

  C[5] = (-2837835.0*H[6] - 9168390.0*H[5] - 16372125.0*H[4] - 18918900*H[3]

          -10135125.0*H[2] + 13783770.0*H[1] + 43648605.0

          -T[4]*(3044160.0*H[2] + 5588352.0*H[1] + 4213440.0)

          +sgn*T[2]*(5556600.0*H[4] + 14817600.0*H[3] + 20790000.0*H[2]

                     + 17297280.0*H[1] + 5405400.0 + 323072.0*T[4]

                     )

          ) / (4194304.0*T[6]);

  C[6] = 0.0;

  *V0 = C[0] + (-4.0*C[3]/T[1]+C[4])/V[4]

             + (-192.0*C[5]/T[3]+144.0*C[6]/T[2])/V[8]

             + sgn * (-C[2]/V[2]

                      + (-24.0*C[4]/T[2]+12.0*C[5]/T[1]-C[6])/V[6] 

                      );

  *V1 = C[1]/V[1] + (8.0*(C[3]/T[2]-C[4]/T[1])+C[5])/V[5]

                  + (384.0*C[5]/T[4] - 768.0*C[6]/T[3])/V[9]

                  + sgn * (Cm1*V[1] + (2.0*C[2]/T[1]-C[3])/V[3]

                           + (48.0*C[4]/T[3]-72.0*C[5]/T[2] + 18.0*C[6]/T[1])/V[7]

                           );

  return GSL_SUCCESS;

}

/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/

/* P^0_{-1/2 + I lambda}

 */

int

gsl_sf_conicalP_0_e(const double lambda, const double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(x <= -1.0) {

    DOMAIN_ERROR(result);

  }

  else if(x == 1.0) {

    result->val = 1.0;

    result->err = 0.0;

    return GSL_SUCCESS;

  }

  else if(lambda == 0.0) {

    gsl_sf_result K;

    int stat_K;

    if(x < 1.0) {

      const double th = acos(x);

      const double s  = sin(0.5*th);

      stat_K = gsl_sf_ellint_Kcomp_e(s, GSL_MODE_DEFAULT, &K);

      result->val  = 2.0/M_PI * K.val;

      result->err  = 2.0/M_PI * K.err;

      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

      return stat_K;

    }

    else {

      const double xi = acosh(x);

      const double c  = cosh(0.5*xi);

      const double t  = tanh(0.5*xi);

      stat_K = gsl_sf_ellint_Kcomp_e(t, GSL_MODE_DEFAULT, &K);

      result->val  = 2.0/M_PI / c * K.val;

      result->err  = 2.0/M_PI / c * K.err;

      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

      return stat_K;

    }

  }

  else if(   (x <= 0.0 && lambda < 1000.0)

          || (x <  0.1 && lambda < 17.0)

          || (x <  0.2 && lambda < 5.0 )

    ) {

    return conicalP_xlt1_hyperg_A(0.0, lambda, x, result);

  }

  else if(   (x <= 0.2 && lambda < 17.0)

          || (x <= 1.5 && lambda < 20.0)

    ) {

    return gsl_sf_hyperg_2F1_conj_e(0.5, lambda, 1.0, (1.0-x)/2, result);

  }

  else if(1.5 < x && lambda < GSL_MAX(x,20.0)) {

    gsl_sf_result P;

    double lm;

    int stat_P = gsl_sf_conicalP_large_x_e(0.0, lambda, x,

                                              &P, &lm

                                              );

    int stat_e = gsl_sf_exp_mult_err_e(lm, 2.0*GSL_DBL_EPSILON * fabs(lm),

                                          P.val, P.err,

                                          result);

    return GSL_ERROR_SELECT_2(stat_e, stat_P);

  }

  else {

    double V0, V1;

    if(x < 1.0) {

      double th  = acos(x);

      double sth = sqrt(1.0-x*x);  /* sin(th) */

      gsl_sf_result I0, I1;

      int stat_I0 = gsl_sf_bessel_I0_scaled_e(th * lambda, &I0;;

      int stat_I1 = gsl_sf_bessel_I1_scaled_e(th * lambda, &I1;;

      int stat_I  = GSL_ERROR_SELECT_2(stat_I0, stat_I1);

      int stat_V  = conicalP_0_V(th, x/sth, lambda, -1.0, &V0, &V1;;

      double bessterm = V0 * I0.val + V1 * I1.val;

      double besserr  = fabs(V0) * I0.err + fabs(V1) * I1.err;

      double arg1 = th*lambda;

      double sqts = sqrt(th/sth);

      int stat_e = gsl_sf_exp_mult_err_e(arg1, 4.0 * GSL_DBL_EPSILON * fabs(arg1),

                                            sqts * bessterm, sqts * besserr,

                                            result);

      return GSL_ERROR_SELECT_3(stat_e, stat_V, stat_I);

    }

    else {

      double sh = sqrt(x-1.0)*sqrt(x+1.0);  /* sinh(xi)      */

      double xi = log(x + sh);              /* xi = acosh(x) */

      gsl_sf_result J0, J1;

      int stat_J0 = gsl_sf_bessel_J0_e(xi * lambda, &J0;;

      int stat_J1 = gsl_sf_bessel_J1_e(xi * lambda, &J1;;

      int stat_J  = GSL_ERROR_SELECT_2(stat_J0, stat_J1);

      int stat_V  = conicalP_0_V(xi, x/sh, lambda, 1.0, &V0, &V1;;

      double bessterm = V0 * J0.val + V1 * J1.val;

      double besserr  = fabs(V0) * J0.err + fabs(V1) * J1.err;

      double pre_val = sqrt(xi/sh);

      double pre_err = 2.0 * fabs(pre_val);

      result->val  = pre_val * bessterm;

      result->err  = pre_val * besserr;

      result->err += pre_err * fabs(bessterm);

      result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

      return GSL_ERROR_SELECT_2(stat_V, stat_J);

    }

  }

}

/* P^1_{-1/2 + I lambda}

 */

int

gsl_sf_conicalP_1_e(const double lambda, const double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(x <= -1.0) {

    DOMAIN_ERROR(result);

  }

  else if(lambda == 0.0) {

    gsl_sf_result K, E;

    int stat_K, stat_E;

    if(x == 1.0) {

      result->val = 0.0;

      result->err = 0.0;

      return GSL_SUCCESS;