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1 // Special functions -*- C++ -*-
3 // Copyright (C) 2006-2018 Free Software Foundation, Inc.
4 //
5 // This file is part of the GNU ISO C++ Library. This library is free
6 // software; you can redistribute it and/or modify it under the
7 // terms of the GNU General Public License as published by the
8 // Free Software Foundation; either version 3, or (at your option)
9 // any later version.
10 //
11 // This library is distributed in the hope that it will be useful,
12 // but WITHOUT ANY WARRANTY; without even the implied warranty of
13 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14 // GNU General Public License for more details.
15 //
16 // Under Section 7 of GPL version 3, you are granted additional
17 // permissions described in the GCC Runtime Library Exception, version
18 // 3.1, as published by the Free Software Foundation.
20 // You should have received a copy of the GNU General Public License and
21 // a copy of the GCC Runtime Library Exception along with this program;
22 // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
23 // <http://www.gnu.org/licenses/>.
25 /** @file tr1/bessel_function.tcc
26 * This is an internal header file, included by other library headers.
27 * Do not attempt to use it directly. @headername{tr1/cmath}
28 */
30 //
31 // ISO C++ 14882 TR1: 5.2 Special functions
32 //
34 // Written by Edward Smith-Rowland.
35 //
36 // References:
37 // (1) Handbook of Mathematical Functions,
38 // ed. Milton Abramowitz and Irene A. Stegun,
39 // Dover Publications,
40 // Section 9, pp. 355-434, Section 10 pp. 435-478
41 // (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
42 // (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
43 // W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
44 // 2nd ed, pp. 240-245
46 #ifndef _GLIBCXX_TR1_BESSEL_FUNCTION_TCC
47 #define _GLIBCXX_TR1_BESSEL_FUNCTION_TCC 1
49 #include "special_function_util.h"
51 namespace std _GLIBCXX_VISIBILITY(default)
52 {
53 _GLIBCXX_BEGIN_NAMESPACE_VERSION
55 #if _GLIBCXX_USE_STD_SPEC_FUNCS
56 # define _GLIBCXX_MATH_NS ::std
57 #elif defined(_GLIBCXX_TR1_CMATH)
58 namespace tr1
59 {
60 # define _GLIBCXX_MATH_NS ::std::tr1
61 #else
62 # error do not include this header directly, use <cmath> or <tr1/cmath>
63 #endif
64 // [5.2] Special functions
66 // Implementation-space details.
67 namespace __detail
68 {
69 /**
70 * @brief Compute the gamma functions required by the Temme series
71 * expansions of @f$ N_\nu(x) @f$ and @f$ K_\nu(x) @f$.
72 * @f[
73 * \Gamma_1 = \frac{1}{2\mu}
74 * [\frac{1}{\Gamma(1 - \mu)} - \frac{1}{\Gamma(1 + \mu)}]
75 * @f]
76 * and
77 * @f[
78 * \Gamma_2 = \frac{1}{2}
79 * [\frac{1}{\Gamma(1 - \mu)} + \frac{1}{\Gamma(1 + \mu)}]
80 * @f]
81 * where @f$ -1/2 <= \mu <= 1/2 @f$ is @f$ \mu = \nu - N @f$ and @f$ N @f$.
82 * is the nearest integer to @f$ \nu @f$.
83 * The values of \f$ \Gamma(1 + \mu) \f$ and \f$ \Gamma(1 - \mu) \f$
84 * are returned as well.
85 *
86 * The accuracy requirements on this are exquisite.
87 *
88 * @param __mu The input parameter of the gamma functions.
89 * @param __gam1 The output function \f$ \Gamma_1(\mu) \f$
90 * @param __gam2 The output function \f$ \Gamma_2(\mu) \f$
91 * @param __gampl The output function \f$ \Gamma(1 + \mu) \f$
92 * @param __gammi The output function \f$ \Gamma(1 - \mu) \f$
93 */
94 template <typename _Tp>
95 void
96 __gamma_temme(_Tp __mu,
97 _Tp & __gam1, _Tp & __gam2, _Tp & __gampl, _Tp & __gammi)
98 {
99 #if _GLIBCXX_USE_C99_MATH_TR1
100 __gampl = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) + __mu);
101 __gammi = _Tp(1) / _GLIBCXX_MATH_NS::tgamma(_Tp(1) - __mu);
102 #else
103 __gampl = _Tp(1) / __gamma(_Tp(1) + __mu);
104 __gammi = _Tp(1) / __gamma(_Tp(1) - __mu);
105 #endif
107 if (std::abs(__mu) < std::numeric_limits<_Tp>::epsilon())
108 __gam1 = -_Tp(__numeric_constants<_Tp>::__gamma_e());
109 else
110 __gam1 = (__gammi - __gampl) / (_Tp(2) * __mu);
112 __gam2 = (__gammi + __gampl) / (_Tp(2));
114 return;
115 }
118 /**
119 * @brief Compute the Bessel @f$ J_\nu(x) @f$ and Neumann
120 * @f$ N_\nu(x) @f$ functions and their first derivatives
121 * @f$ J'_\nu(x) @f$ and @f$ N'_\nu(x) @f$ respectively.
122 * These four functions are computed together for numerical
123 * stability.
124 *
125 * @param __nu The order of the Bessel functions.
126 * @param __x The argument of the Bessel functions.
127 * @param __Jnu The output Bessel function of the first kind.
128 * @param __Nnu The output Neumann function (Bessel function of the second kind).
129 * @param __Jpnu The output derivative of the Bessel function of the first kind.
130 * @param __Npnu The output derivative of the Neumann function.
131 */
132 template <typename _Tp>
133 void
134 __bessel_jn(_Tp __nu, _Tp __x,
135 _Tp & __Jnu, _Tp & __Nnu, _Tp & __Jpnu, _Tp & __Npnu)
136 {
137 if (__x == _Tp(0))
138 {
139 if (__nu == _Tp(0))
140 {
141 __Jnu = _Tp(1);
142 __Jpnu = _Tp(0);
143 }
144 else if (__nu == _Tp(1))
145 {
146 __Jnu = _Tp(0);
147 __Jpnu = _Tp(0.5L);
148 }
149 else
150 {
151 __Jnu = _Tp(0);
152 __Jpnu = _Tp(0);
153 }
154 __Nnu = -std::numeric_limits<_Tp>::infinity();
155 __Npnu = std::numeric_limits<_Tp>::infinity();
156 return;
157 }
159 const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
160 // When the multiplier is N i.e.
161 // fp_min = N * min()
162 // Then J_0 and N_0 tank at x = 8 * N (J_0 = 0 and N_0 = nan)!
163 //const _Tp __fp_min = _Tp(20) * std::numeric_limits<_Tp>::min();
164 const _Tp __fp_min = std::sqrt(std::numeric_limits<_Tp>::min());
165 const int __max_iter = 15000;
166 const _Tp __x_min = _Tp(2);
168 const int __nl = (__x < __x_min
169 ? static_cast<int>(__nu + _Tp(0.5L))
170 : std::max(0, static_cast<int>(__nu - __x + _Tp(1.5L))));
172 const _Tp __mu = __nu - __nl;
173 const _Tp __mu2 = __mu * __mu;
174 const _Tp __xi = _Tp(1) / __x;
175 const _Tp __xi2 = _Tp(2) * __xi;
176 _Tp __w = __xi2 / __numeric_constants<_Tp>::__pi();
177 int __isign = 1;
178 _Tp __h = __nu * __xi;
179 if (__h < __fp_min)
180 __h = __fp_min;
181 _Tp __b = __xi2 * __nu;
182 _Tp __d = _Tp(0);
183 _Tp __c = __h;
184 int __i;
185 for (__i = 1; __i <= __max_iter; ++__i)
186 {
187 __b += __xi2;
188 __d = __b - __d;
189 if (std::abs(__d) < __fp_min)
190 __d = __fp_min;
191 __c = __b - _Tp(1) / __c;
192 if (std::abs(__c) < __fp_min)
193 __c = __fp_min;
194 __d = _Tp(1) / __d;
195 const _Tp __del = __c * __d;
196 __h *= __del;
197 if (__d < _Tp(0))
198 __isign = -__isign;
199 if (std::abs(__del - _Tp(1)) < __eps)
200 break;
201 }
202 if (__i > __max_iter)
203 std::__throw_runtime_error(__N("Argument x too large in __bessel_jn; "
204 "try asymptotic expansion."));
205 _Tp __Jnul = __isign * __fp_min;
206 _Tp __Jpnul = __h * __Jnul;
207 _Tp __Jnul1 = __Jnul;
208 _Tp __Jpnu1 = __Jpnul;
209 _Tp __fact = __nu * __xi;
210 for ( int __l = __nl; __l >= 1; --__l )
211 {
212 const _Tp __Jnutemp = __fact * __Jnul + __Jpnul;
213 __fact -= __xi;
214 __Jpnul = __fact * __Jnutemp - __Jnul;
215 __Jnul = __Jnutemp;
216 }
217 if (__Jnul == _Tp(0))
218 __Jnul = __eps;
219 _Tp __f= __Jpnul / __Jnul;
220 _Tp __Nmu, __Nnu1, __Npmu, __Jmu;
221 if (__x < __x_min)
222 {
223 const _Tp __x2 = __x / _Tp(2);
224 const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu;
225 _Tp __fact = (std::abs(__pimu) < __eps
226 ? _Tp(1) : __pimu / std::sin(__pimu));
227 _Tp __d = -std::log(__x2);
228 _Tp __e = __mu * __d;
229 _Tp __fact2 = (std::abs(__e) < __eps
230 ? _Tp(1) : std::sinh(__e) / __e);
231 _Tp __gam1, __gam2, __gampl, __gammi;
232 __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi);
233 _Tp __ff = (_Tp(2) / __numeric_constants<_Tp>::__pi())
234 * __fact * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d);
235 __e = std::exp(__e);
236 _Tp __p = __e / (__numeric_constants<_Tp>::__pi() * __gampl);
237 _Tp __q = _Tp(1) / (__e * __numeric_constants<_Tp>::__pi() * __gammi);
238 const _Tp __pimu2 = __pimu / _Tp(2);
239 _Tp __fact3 = (std::abs(__pimu2) < __eps
240 ? _Tp(1) : std::sin(__pimu2) / __pimu2 );
241 _Tp __r = __numeric_constants<_Tp>::__pi() * __pimu2 * __fact3 * __fact3;
242 _Tp __c = _Tp(1);
243 __d = -__x2 * __x2;
244 _Tp __sum = __ff + __r * __q;
245 _Tp __sum1 = __p;
246 for (__i = 1; __i <= __max_iter; ++__i)
247 {
248 __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2);
249 __c *= __d / _Tp(__i);
250 __p /= _Tp(__i) - __mu;
251 __q /= _Tp(__i) + __mu;
252 const _Tp __del = __c * (__ff + __r * __q);
253 __sum += __del;
254 const _Tp __del1 = __c * __p - __i * __del;
255 __sum1 += __del1;
256 if ( std::abs(__del) < __eps * (_Tp(1) + std::abs(__sum)) )
257 break;
258 }
259 if ( __i > __max_iter )
260 std::__throw_runtime_error(__N("Bessel y series failed to converge "
261 "in __bessel_jn."));
262 __Nmu = -__sum;
263 __Nnu1 = -__sum1 * __xi2;
264 __Npmu = __mu * __xi * __Nmu - __Nnu1;
265 __Jmu = __w / (__Npmu - __f * __Nmu);
266 }
267 else
268 {
269 _Tp __a = _Tp(0.25L) - __mu2;
270 _Tp __q = _Tp(1);
271 _Tp __p = -__xi / _Tp(2);
272 _Tp __br = _Tp(2) * __x;
273 _Tp __bi = _Tp(2);
274 _Tp __fact = __a * __xi / (__p * __p + __q * __q);
275 _Tp __cr = __br + __q * __fact;
276 _Tp __ci = __bi + __p * __fact;
277 _Tp __den = __br * __br + __bi * __bi;
278 _Tp __dr = __br / __den;
279 _Tp __di = -__bi / __den;
280 _Tp __dlr = __cr * __dr - __ci * __di;
281 _Tp __dli = __cr * __di + __ci * __dr;
282 _Tp __temp = __p * __dlr - __q * __dli;
283 __q = __p * __dli + __q * __dlr;
284 __p = __temp;
285 int __i;
286 for (__i = 2; __i <= __max_iter; ++__i)
287 {
288 __a += _Tp(2 * (__i - 1));
289 __bi += _Tp(2);
290 __dr = __a * __dr + __br;
291 __di = __a * __di + __bi;
292 if (std::abs(__dr) + std::abs(__di) < __fp_min)
293 __dr = __fp_min;
294 __fact = __a / (__cr * __cr + __ci * __ci);
295 __cr = __br + __cr * __fact;
296 __ci = __bi - __ci * __fact;
297 if (std::abs(__cr) + std::abs(__ci) < __fp_min)
298 __cr = __fp_min;
299 __den = __dr * __dr + __di * __di;
300 __dr /= __den;
301 __di /= -__den;
302 __dlr = __cr * __dr - __ci * __di;
303 __dli = __cr * __di + __ci * __dr;
304 __temp = __p * __dlr - __q * __dli;
305 __q = __p * __dli + __q * __dlr;
306 __p = __temp;
307 if (std::abs(__dlr - _Tp(1)) + std::abs(__dli) < __eps)
308 break;
309 }
310 if (__i > __max_iter)
311 std::__throw_runtime_error(__N("Lentz's method failed "
312 "in __bessel_jn."));
313 const _Tp __gam = (__p - __f) / __q;
314 __Jmu = std::sqrt(__w / ((__p - __f) * __gam + __q));
315 #if _GLIBCXX_USE_C99_MATH_TR1
316 __Jmu = _GLIBCXX_MATH_NS::copysign(__Jmu, __Jnul);
317 #else
318 if (__Jmu * __Jnul < _Tp(0))
319 __Jmu = -__Jmu;
320 #endif
321 __Nmu = __gam * __Jmu;
322 __Npmu = (__p + __q / __gam) * __Nmu;
323 __Nnu1 = __mu * __xi * __Nmu - __Npmu;
324 }
325 __fact = __Jmu / __Jnul;
326 __Jnu = __fact * __Jnul1;
327 __Jpnu = __fact * __Jpnu1;
328 for (__i = 1; __i <= __nl; ++__i)
329 {
330 const _Tp __Nnutemp = (__mu + __i) * __xi2 * __Nnu1 - __Nmu;
331 __Nmu = __Nnu1;
332 __Nnu1 = __Nnutemp;
333 }
334 __Nnu = __Nmu;
335 __Npnu = __nu * __xi * __Nmu - __Nnu1;
337 return;
338 }
341 /**
342 * @brief This routine computes the asymptotic cylindrical Bessel
343 * and Neumann functions of order nu: \f$ J_{\nu} \f$,
344 * \f$ N_{\nu} \f$.
345 *
346 * References:
347 * (1) Handbook of Mathematical Functions,
348 * ed. Milton Abramowitz and Irene A. Stegun,
349 * Dover Publications,
350 * Section 9 p. 364, Equations 9.2.5-9.2.10
351 *
352 * @param __nu The order of the Bessel functions.
353 * @param __x The argument of the Bessel functions.
354 * @param __Jnu The output Bessel function of the first kind.
355 * @param __Nnu The output Neumann function (Bessel function of the second kind).
356 */
357 template <typename _Tp>
358 void
359 __cyl_bessel_jn_asymp(_Tp __nu, _Tp __x, _Tp & __Jnu, _Tp & __Nnu)
360 {
361 const _Tp __mu = _Tp(4) * __nu * __nu;
362 const _Tp __mum1 = __mu - _Tp(1);
363 const _Tp __mum9 = __mu - _Tp(9);
364 const _Tp __mum25 = __mu - _Tp(25);
365 const _Tp __mum49 = __mu - _Tp(49);
366 const _Tp __xx = _Tp(64) * __x * __x;
367 const _Tp __P = _Tp(1) - __mum1 * __mum9 / (_Tp(2) * __xx)
368 * (_Tp(1) - __mum25 * __mum49 / (_Tp(12) * __xx));
369 const _Tp __Q = __mum1 / (_Tp(8) * __x)
370 * (_Tp(1) - __mum9 * __mum25 / (_Tp(6) * __xx));
372 const _Tp __chi = __x - (__nu + _Tp(0.5L))
373 * __numeric_constants<_Tp>::__pi_2();
374 const _Tp __c = std::cos(__chi);
375 const _Tp __s = std::sin(__chi);
377 const _Tp __coef = std::sqrt(_Tp(2)
378 / (__numeric_constants<_Tp>::__pi() * __x));
379 __Jnu = __coef * (__c * __P - __s * __Q);
380 __Nnu = __coef * (__s * __P + __c * __Q);
382 return;
383 }
386 /**
387 * @brief This routine returns the cylindrical Bessel functions
388 * of order \f$ \nu \f$: \f$ J_{\nu} \f$ or \f$ I_{\nu} \f$
389 * by series expansion.
390 *
391 * The modified cylindrical Bessel function is:
392 * @f[
393 * Z_{\nu}(x) = \sum_{k=0}^{\infty}
394 * \frac{\sigma^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
395 * @f]
396 * where \f$ \sigma = +1 \f$ or\f$ -1 \f$ for
397 * \f$ Z = I \f$ or \f$ J \f$ respectively.
398 *
399 * See Abramowitz & Stegun, 9.1.10
400 * Abramowitz & Stegun, 9.6.7
401 * (1) Handbook of Mathematical Functions,
402 * ed. Milton Abramowitz and Irene A. Stegun,
403 * Dover Publications,
404 * Equation 9.1.10 p. 360 and Equation 9.6.10 p. 375
405 *
406 * @param __nu The order of the Bessel function.
407 * @param __x The argument of the Bessel function.
408 * @param __sgn The sign of the alternate terms
409 * -1 for the Bessel function of the first kind.
410 * +1 for the modified Bessel function of the first kind.
411 * @return The output Bessel function.
412 */
413 template <typename _Tp>
414 _Tp
415 __cyl_bessel_ij_series(_Tp __nu, _Tp __x, _Tp __sgn,
416 unsigned int __max_iter)
417 {
418 if (__x == _Tp(0))
419 return __nu == _Tp(0) ? _Tp(1) : _Tp(0);
421 const _Tp __x2 = __x / _Tp(2);
422 _Tp __fact = __nu * std::log(__x2);
423 #if _GLIBCXX_USE_C99_MATH_TR1
424 __fact -= _GLIBCXX_MATH_NS::lgamma(__nu + _Tp(1));
425 #else
426 __fact -= __log_gamma(__nu + _Tp(1));
427 #endif
428 __fact = std::exp(__fact);
429 const _Tp __xx4 = __sgn * __x2 * __x2;
430 _Tp __Jn = _Tp(1);
431 _Tp __term = _Tp(1);
433 for (unsigned int __i = 1; __i < __max_iter; ++__i)
434 {
435 __term *= __xx4 / (_Tp(__i) * (__nu + _Tp(__i)));
436 __Jn += __term;
437 if (std::abs(__term / __Jn) < std::numeric_limits<_Tp>::epsilon())
438 break;
439 }
441 return __fact * __Jn;
442 }
445 /**
446 * @brief Return the Bessel function of order \f$ \nu \f$:
447 * \f$ J_{\nu}(x) \f$.
448 *
449 * The cylindrical Bessel function is:
450 * @f[
451 * J_{\nu}(x) = \sum_{k=0}^{\infty}
452 * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}
453 * @f]
454 *
455 * @param __nu The order of the Bessel function.
456 * @param __x The argument of the Bessel function.
457 * @return The output Bessel function.
458 */
459 template<typename _Tp>
460 _Tp
461 __cyl_bessel_j(_Tp __nu, _Tp __x)
462 {
463 if (__nu < _Tp(0) || __x < _Tp(0))
464 std::__throw_domain_error(__N("Bad argument "
465 "in __cyl_bessel_j."));
466 else if (__isnan(__nu) || __isnan(__x))
467 return std::numeric_limits<_Tp>::quiet_NaN();
468 else if (__x * __x < _Tp(10) * (__nu + _Tp(1)))
469 return __cyl_bessel_ij_series(__nu, __x, -_Tp(1), 200);
470 else if (__x > _Tp(1000))
471 {
472 _Tp __J_nu, __N_nu;
473 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
474 return __J_nu;
475 }
476 else
477 {
478 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
479 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
480 return __J_nu;
481 }
482 }
485 /**
486 * @brief Return the Neumann function of order \f$ \nu \f$:
487 * \f$ N_{\nu}(x) \f$.
488 *
489 * The Neumann function is defined by:
490 * @f[
491 * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)}
492 * {\sin \nu\pi}
493 * @f]
494 * where for integral \f$ \nu = n \f$ a limit is taken:
495 * \f$ lim_{\nu \to n} \f$.
496 *
497 * @param __nu The order of the Neumann function.
498 * @param __x The argument of the Neumann function.
499 * @return The output Neumann function.
500 */
501 template<typename _Tp>
502 _Tp
503 __cyl_neumann_n(_Tp __nu, _Tp __x)
504 {
505 if (__nu < _Tp(0) || __x < _Tp(0))
506 std::__throw_domain_error(__N("Bad argument "
507 "in __cyl_neumann_n."));
508 else if (__isnan(__nu) || __isnan(__x))
509 return std::numeric_limits<_Tp>::quiet_NaN();
510 else if (__x > _Tp(1000))
511 {
512 _Tp __J_nu, __N_nu;
513 __cyl_bessel_jn_asymp(__nu, __x, __J_nu, __N_nu);
514 return __N_nu;
515 }
516 else
517 {
518 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
519 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
520 return __N_nu;
521 }
522 }
525 /**
526 * @brief Compute the spherical Bessel @f$ j_n(x) @f$
527 * and Neumann @f$ n_n(x) @f$ functions and their first
528 * derivatives @f$ j'_n(x) @f$ and @f$ n'_n(x) @f$
529 * respectively.
530 *
531 * @param __n The order of the spherical Bessel function.
532 * @param __x The argument of the spherical Bessel function.
533 * @param __j_n The output spherical Bessel function.
534 * @param __n_n The output spherical Neumann function.
535 * @param __jp_n The output derivative of the spherical Bessel function.
536 * @param __np_n The output derivative of the spherical Neumann function.
537 */
538 template <typename _Tp>
539 void
540 __sph_bessel_jn(unsigned int __n, _Tp __x,
541 _Tp & __j_n, _Tp & __n_n, _Tp & __jp_n, _Tp & __np_n)
542 {
543 const _Tp __nu = _Tp(__n) + _Tp(0.5L);
545 _Tp __J_nu, __N_nu, __Jp_nu, __Np_nu;
546 __bessel_jn(__nu, __x, __J_nu, __N_nu, __Jp_nu, __Np_nu);
548 const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2()
549 / std::sqrt(__x);
551 __j_n = __factor * __J_nu;
552 __n_n = __factor * __N_nu;
553 __jp_n = __factor * __Jp_nu - __j_n / (_Tp(2) * __x);
554 __np_n = __factor * __Np_nu - __n_n / (_Tp(2) * __x);
556 return;
557 }
560 /**
561 * @brief Return the spherical Bessel function
562 * @f$ j_n(x) @f$ of order n.
563 *
564 * The spherical Bessel function is defined by:
565 * @f[
566 * j_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x)
567 * @f]
568 *
569 * @param __n The order of the spherical Bessel function.
570 * @param __x The argument of the spherical Bessel function.
571 * @return The output spherical Bessel function.
572 */
573 template <typename _Tp>
574 _Tp
575 __sph_bessel(unsigned int __n, _Tp __x)
576 {
577 if (__x < _Tp(0))
578 std::__throw_domain_error(__N("Bad argument "
579 "in __sph_bessel."));
580 else if (__isnan(__x))
581 return std::numeric_limits<_Tp>::quiet_NaN();
582 else if (__x == _Tp(0))
583 {
584 if (__n == 0)
585 return _Tp(1);
586 else
587 return _Tp(0);
588 }
589 else
590 {
591 _Tp __j_n, __n_n, __jp_n, __np_n;
592 __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n);
593 return __j_n;
594 }
595 }
598 /**
599 * @brief Return the spherical Neumann function
600 * @f$ n_n(x) @f$.
601 *
602 * The spherical Neumann function is defined by:
603 * @f[
604 * n_n(x) = \left( \frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x)
605 * @f]
606 *
607 * @param __n The order of the spherical Neumann function.
608 * @param __x The argument of the spherical Neumann function.
609 * @return The output spherical Neumann function.
610 */
611 template <typename _Tp>
612 _Tp
613 __sph_neumann(unsigned int __n, _Tp __x)
614 {
615 if (__x < _Tp(0))
616 std::__throw_domain_error(__N("Bad argument "
617 "in __sph_neumann."));
618 else if (__isnan(__x))
619 return std::numeric_limits<_Tp>::quiet_NaN();
620 else if (__x == _Tp(0))
621 return -std::numeric_limits<_Tp>::infinity();
622 else
623 {
624 _Tp __j_n, __n_n, __jp_n, __np_n;
625 __sph_bessel_jn(__n, __x, __j_n, __n_n, __jp_n, __np_n);
626 return __n_n;
627 }
628 }
629 } // namespace __detail
630 #undef _GLIBCXX_MATH_NS
631 #if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
632 } // namespace tr1
633 #endif
635 _GLIBCXX_END_NAMESPACE_VERSION
636 }
638 #endif // _GLIBCXX_TR1_BESSEL_FUNCTION_TCC