/// Log upper Pochhammer symbol.
double
lpochhammer_u(double a, double x)
{
gsl_sf_result result;
int stat = gsl_sf_lnpoch_e(a, x, &result);
if (stat != GSL_SUCCESS)
{
std::ostringstream msg("Error in lpochhammer_u:");
msg << " a=" << a << " x=" << x;
throw std::runtime_error(msg.str());
}
else
return result.val;
}
int
gsl_sf_legendre_sphPlm_e(const int l, int m, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(m < 0 || l < m || x < -1.0 || x > 1.0) {
DOMAIN_ERROR(result);
}
else if(m == 0) {
gsl_sf_result P;
int stat_P = gsl_sf_legendre_Pl_e(l, x, &P);
double pre = sqrt((2.0*l + 1.0)/(4.0*M_PI));
result->val = pre * P.val;
result->err = pre * P.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_P;
}
else if(x == 1.0 || x == -1.0) {
/* m > 0 here */
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
/* m > 0 and |x| < 1 here */
/* Starting value for recursion.
* Y_m^m(x) = sqrt( (2m+1)/(4pi m) gamma(m+1/2)/gamma(m) ) (-1)^m (1-x^2)^(m/2) / pi^(1/4)
*/
gsl_sf_result lncirc;
gsl_sf_result lnpoch;
double lnpre_val;
double lnpre_err;
gsl_sf_result ex_pre;
double sr;
const double sgn = ( GSL_IS_ODD(m) ? -1.0 : 1.0);
const double y_mmp1_factor = x * sqrt(2.0*m + 3.0);
double y_mm, y_mm_err;
double y_mmp1;
gsl_sf_log_1plusx_e(-x*x, &lncirc);
gsl_sf_lnpoch_e(m, 0.5, &lnpoch); /* Gamma(m+1/2)/Gamma(m) */
lnpre_val = -0.25*M_LNPI + 0.5 * (lnpoch.val + m*lncirc.val);
lnpre_err = 0.25*M_LNPI*GSL_DBL_EPSILON + 0.5 * (lnpoch.err + fabs(m)*lncirc.err);
gsl_sf_exp_err_e(lnpre_val, lnpre_err, &ex_pre);
sr = sqrt((2.0+1.0/m)/(4.0*M_PI));
y_mm = sgn * sr * ex_pre.val;
y_mmp1 = y_mmp1_factor * y_mm;
y_mm_err = 2.0 * GSL_DBL_EPSILON * fabs(y_mm) + sr * ex_pre.err;
y_mm_err *= 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-x));
if(l == m){
result->val = y_mm;
result->err = y_mm_err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(y_mm);
return GSL_SUCCESS;
}
else if(l == m + 1) {
result->val = y_mmp1;
result->err = fabs(y_mmp1_factor) * y_mm_err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(y_mmp1);
return GSL_SUCCESS;
}
else{
double y_ell = 0.0;
int ell;
/* Compute Y_l^m, l > m+1, upward recursion on l. */
for(ell=m+2; ell <= l; ell++){
const double rat1 = (double)(ell-m)/(double)(ell+m);
const double rat2 = (ell-m-1.0)/(ell+m-1.0);
const double factor1 = sqrt(rat1*(2*ell+1)*(2*ell-1));
const double factor2 = sqrt(rat1*rat2*(2*ell+1)/(2*ell-3));
y_ell = (x*y_mmp1*factor1 - (ell+m-1)*y_mm*factor2) / (ell-m);
y_mm = y_mmp1;
y_mmp1 = y_ell;
}
result->val = y_ell;
result->err = (0.5*(l-m) + 1.0) * GSL_DBL_EPSILON * fabs(y_ell);
result->err += fabs(y_mm_err/y_mm) * fabs(y_ell);
return GSL_SUCCESS;
}
}
}
int
gsl_sf_legendre_sphPlm_array(const int lmax, int m, const double x, double * result_array)
{
/* CHECK_POINTER(result_array) */
if(m < 0 || lmax < m || x < -1.0 || x > 1.0) {
GSL_ERROR ("error", GSL_EDOM);
}
else if(m > 0 && (x == 1.0 || x == -1.0)) {
int ell;
for(ell=m; ell<=lmax; ell++) result_array[ell-m] = 0.0;
return GSL_SUCCESS;
}
else {
double y_mm;
double y_mmp1;
if(m == 0) {
y_mm = 0.5/M_SQRTPI; /* Y00 = 1/sqrt(4pi) */
y_mmp1 = x * M_SQRT3 * y_mm;
}
else {
/* |x| < 1 here */
gsl_sf_result lncirc;
gsl_sf_result lnpoch;
double lnpre;
const double sgn = ( GSL_IS_ODD(m) ? -1.0 : 1.0);
gsl_sf_log_1plusx_e(-x*x, &lncirc);
gsl_sf_lnpoch_e(m, 0.5, &lnpoch); /* Gamma(m+1/2)/Gamma(m) */
lnpre = -0.25*M_LNPI + 0.5 * (lnpoch.val + m*lncirc.val);
y_mm = sqrt((2.0+1.0/m)/(4.0*M_PI)) * sgn * exp(lnpre);
y_mmp1 = x * sqrt(2.0*m + 3.0) * y_mm;
}
if(lmax == m){
result_array[0] = y_mm;
return GSL_SUCCESS;
}
else if(lmax == m + 1) {
result_array[0] = y_mm;
result_array[1] = y_mmp1;
return GSL_SUCCESS;
}
else{
double y_ell;
int ell;
result_array[0] = y_mm;
result_array[1] = y_mmp1;
/* Compute Y_l^m, l > m+1, upward recursion on l. */
for(ell=m+2; ell <= lmax; ell++){
const double rat1 = (double)(ell-m)/(double)(ell+m);
const double rat2 = (ell-m-1.0)/(ell+m-1.0);
const double factor1 = sqrt(rat1*(2*ell+1)*(2*ell-1));
const double factor2 = sqrt(rat1*rat2*(2*ell+1)/(2*ell-3));
y_ell = (x*y_mmp1*factor1 - (ell+m-1)*y_mm*factor2) / (ell-m);
y_mm = y_mmp1;
y_mmp1 = y_ell;
result_array[ell-m] = y_ell;
}
}
return GSL_SUCCESS;
}
}
double gsl_sf_lnpoch(const double a, const double x)
{
EVAL_RESULT(gsl_sf_lnpoch_e(a, x, &result));
}
