int
gsl_sf_bessel_Jn_array(int nmin, int nmax, double x, double * result_array)
{
/* CHECK_POINTER(result_array) */
if(nmin < 0 || nmax < nmin) {
int n;
for(n=nmax; n>=nmin; n--) {
result_array[n-nmin] = 0.0;
}
GSL_ERROR ("domain error", GSL_EDOM);
}
else if(x == 0.0) {
int n;
for(n=nmax; n>=nmin; n--) {
result_array[n-nmin] = 0.0;
}
if(nmin == 0) result_array[0] = 1.0;
return GSL_SUCCESS;
}
else {
gsl_sf_result r_Jnp1;
gsl_sf_result r_Jn;
int stat_np1 = gsl_sf_bessel_Jn_e(nmax+1, x, &r_Jnp1);
int stat_n = gsl_sf_bessel_Jn_e(nmax, x, &r_Jn);
int stat = GSL_ERROR_SELECT_2(stat_np1, stat_n);
double Jnp1 = r_Jnp1.val;
double Jn = r_Jn.val;
double Jnm1;
int n;
if(stat == GSL_SUCCESS) {
for(n=nmax; n>=nmin; n--) {
result_array[n-nmin] = Jn;
Jnm1 = -Jnp1 + 2.0*n/x * Jn;
Jnp1 = Jn;
Jn = Jnm1;
}
}
else {
for(n=nmax; n>=nmin; n--) {
result_array[n-nmin] = 0.0;
}
}
return stat;
}
}
int
gsl_sf_legendre_Pl_e(const int l, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(l < 0 || x < -1.0 || x > 1.0) {
DOMAIN_ERROR(result);
}
else if(l == 0) {
result->val = 1.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(l == 1) {
result->val = x;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(l == 2) {
result->val = 0.5 * (3.0*x*x - 1.0);
result->err = 3.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x == 1.0) {
result->val = 1.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(x == -1.0) {
result->val = ( GSL_IS_ODD(l) ? -1.0 : 1.0 );
result->err = 0.0;
return GSL_SUCCESS;
}
else if(l < 100000) {
/* Compute by upward recurrence on l.
*/
double p_mm = 1.0; /* P_0(x) */
double p_mmp1 = x; /* P_1(x) */
double p_ell = p_mmp1;
int ell;
for(ell=2; ell <= l; ell++){
p_ell = (x*(2*ell-1)*p_mmp1 - (ell-1)*p_mm) / ell;
p_mm = p_mmp1;
p_mmp1 = p_ell;
}
result->val = p_ell;
result->err = (0.5 * ell + 1.0) * GSL_DBL_EPSILON * fabs(p_ell);
return GSL_SUCCESS;
}
else {
/* Asymptotic expansion.
* [Olver, p. 473]
*/
double u = l + 0.5;
double th = acos(x);
gsl_sf_result J0;
gsl_sf_result Jm1;
int stat_J0 = gsl_sf_bessel_J0_e(u*th, &J0);
int stat_Jm1 = gsl_sf_bessel_Jn_e(-1, u*th, &Jm1);
double pre;
double B00;
double c1;
/* B00 = 1/8 (1 - th cot(th) / th^2
* pre = sqrt(th/sin(th))
*/
if(th < GSL_ROOT4_DBL_EPSILON) {
B00 = (1.0 + th*th/15.0)/24.0;
pre = 1.0 + th*th/12.0;
}
else {
double sin_th = sqrt(1.0 - x*x);
double cot_th = x / sin_th;
B00 = 1.0/8.0 * (1.0 - th * cot_th) / (th*th);
pre = sqrt(th/sin_th);
}
c1 = th/u * B00;
result->val = pre * (J0.val + c1 * Jm1.val);
result->err = pre * (J0.err + fabs(c1) * Jm1.err);
result->err += GSL_SQRT_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_J0, stat_Jm1);
}
}
double gsl_sf_bessel_Jn(const int n, const double x)
{
EVAL_RESULT(gsl_sf_bessel_Jn_e(n, x, &result));
}
