1 | // Mathematical Special Functions for -*- C++ -*- |
2 | |
3 | // Copyright (C) 2006-2021 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. |
19 | |
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/>. |
24 | |
25 | /** @file bits/specfun.h |
26 | * This is an internal header file, included by other library headers. |
27 | * Do not attempt to use it directly. @headername{cmath} |
28 | */ |
29 | |
30 | #ifndef _GLIBCXX_BITS_SPECFUN_H |
31 | #define _GLIBCXX_BITS_SPECFUN_H 1 |
32 | |
33 | #pragma GCC visibility push(default) |
34 | |
35 | #include <bits/c++config.h> |
36 | |
37 | #define __STDCPP_MATH_SPEC_FUNCS__ 201003L |
38 | |
39 | #define __cpp_lib_math_special_functions 201603L |
40 | |
41 | #if __cplusplus <= 201403L && __STDCPP_WANT_MATH_SPEC_FUNCS__ == 0 |
42 | # error include <cmath> and define __STDCPP_WANT_MATH_SPEC_FUNCS__ |
43 | #endif |
44 | |
45 | #include <bits/stl_algobase.h> |
46 | #include <limits> |
47 | #include <type_traits> |
48 | |
49 | #include <tr1/gamma.tcc> |
50 | #include <tr1/bessel_function.tcc> |
51 | #include <tr1/beta_function.tcc> |
52 | #include <tr1/ell_integral.tcc> |
53 | #include <tr1/exp_integral.tcc> |
54 | #include <tr1/hypergeometric.tcc> |
55 | #include <tr1/legendre_function.tcc> |
56 | #include <tr1/modified_bessel_func.tcc> |
57 | #include <tr1/poly_hermite.tcc> |
58 | #include <tr1/poly_laguerre.tcc> |
59 | #include <tr1/riemann_zeta.tcc> |
60 | |
61 | namespace std _GLIBCXX_VISIBILITY(default) |
62 | { |
63 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
64 | |
65 | /** |
66 | * @defgroup mathsf Mathematical Special Functions |
67 | * @ingroup numerics |
68 | * |
69 | * @section mathsf_desc Mathematical Special Functions |
70 | * |
71 | * A collection of advanced mathematical special functions, |
72 | * defined by ISO/IEC IS 29124 and then added to ISO C++ 2017. |
73 | * |
74 | * |
75 | * @subsection mathsf_intro Introduction and History |
76 | * The first significant library upgrade on the road to C++2011, |
77 | * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2005/n1836.pdf"> |
78 | * TR1</a>, included a set of 23 mathematical functions that significantly |
79 | * extended the standard transcendental functions inherited from C and declared |
80 | * in @<cmath@>. |
81 | * |
82 | * Although most components from TR1 were eventually adopted for C++11 these |
83 | * math functions were left behind out of concern for implementability. |
84 | * The math functions were published as a separate international standard |
85 | * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2010/n3060.pdf"> |
86 | * IS 29124 - Extensions to the C++ Library to Support Mathematical Special |
87 | * Functions</a>. |
88 | * |
89 | * For C++17 these functions were incorporated into the main standard. |
90 | * |
91 | * @subsection mathsf_contents Contents |
92 | * The following functions are implemented in namespace @c std: |
93 | * - @ref assoc_laguerre "assoc_laguerre - Associated Laguerre functions" |
94 | * - @ref assoc_legendre "assoc_legendre - Associated Legendre functions" |
95 | * - @ref beta "beta - Beta functions" |
96 | * - @ref comp_ellint_1 "comp_ellint_1 - Complete elliptic functions of the first kind" |
97 | * - @ref comp_ellint_2 "comp_ellint_2 - Complete elliptic functions of the second kind" |
98 | * - @ref comp_ellint_3 "comp_ellint_3 - Complete elliptic functions of the third kind" |
99 | * - @ref cyl_bessel_i "cyl_bessel_i - Regular modified cylindrical Bessel functions" |
100 | * - @ref cyl_bessel_j "cyl_bessel_j - Cylindrical Bessel functions of the first kind" |
101 | * - @ref cyl_bessel_k "cyl_bessel_k - Irregular modified cylindrical Bessel functions" |
102 | * - @ref cyl_neumann "cyl_neumann - Cylindrical Neumann functions or Cylindrical Bessel functions of the second kind" |
103 | * - @ref ellint_1 "ellint_1 - Incomplete elliptic functions of the first kind" |
104 | * - @ref ellint_2 "ellint_2 - Incomplete elliptic functions of the second kind" |
105 | * - @ref ellint_3 "ellint_3 - Incomplete elliptic functions of the third kind" |
106 | * - @ref expint "expint - The exponential integral" |
107 | * - @ref hermite "hermite - Hermite polynomials" |
108 | * - @ref laguerre "laguerre - Laguerre functions" |
109 | * - @ref legendre "legendre - Legendre polynomials" |
110 | * - @ref riemann_zeta "riemann_zeta - The Riemann zeta function" |
111 | * - @ref sph_bessel "sph_bessel - Spherical Bessel functions" |
112 | * - @ref sph_legendre "sph_legendre - Spherical Legendre functions" |
113 | * - @ref sph_neumann "sph_neumann - Spherical Neumann functions" |
114 | * |
115 | * The hypergeometric functions were stricken from the TR29124 and C++17 |
116 | * versions of this math library because of implementation concerns. |
117 | * However, since they were in the TR1 version and since they are popular |
118 | * we kept them as an extension in namespace @c __gnu_cxx: |
119 | * - @ref __gnu_cxx::conf_hyperg "conf_hyperg - Confluent hypergeometric functions" |
120 | * - @ref __gnu_cxx::hyperg "hyperg - Hypergeometric functions" |
121 | * |
122 | * <!-- @subsection mathsf_general General Features --> |
123 | * |
124 | * @subsection mathsf_promotion Argument Promotion |
125 | * The arguments suppled to the non-suffixed functions will be promoted |
126 | * according to the following rules: |
127 | * 1. If any argument intended to be floating point is given an integral value |
128 | * That integral value is promoted to double. |
129 | * 2. All floating point arguments are promoted up to the largest floating |
130 | * point precision among them. |
131 | * |
132 | * @subsection mathsf_NaN NaN Arguments |
133 | * If any of the floating point arguments supplied to these functions is |
134 | * invalid or NaN (std::numeric_limits<Tp>::quiet_NaN), |
135 | * the value NaN is returned. |
136 | * |
137 | * @subsection mathsf_impl Implementation |
138 | * |
139 | * We strive to implement the underlying math with type generic algorithms |
140 | * to the greatest extent possible. In practice, the functions are thin |
141 | * wrappers that dispatch to function templates. Type dependence is |
142 | * controlled with std::numeric_limits and functions thereof. |
143 | * |
144 | * We don't promote @c float to @c double or @c double to <tt>long double</tt> |
145 | * reflexively. The goal is for @c float functions to operate more quickly, |
146 | * at the cost of @c float accuracy and possibly a smaller domain of validity. |
147 | * Similaryly, <tt>long double</tt> should give you more dynamic range |
148 | * and slightly more pecision than @c double on many systems. |
149 | * |
150 | * @subsection mathsf_testing Testing |
151 | * |
152 | * These functions have been tested against equivalent implementations |
153 | * from the <a href="http://www.gnu.org/software/gsl"> |
154 | * Gnu Scientific Library, GSL</a> and |
155 | * <a href="http://www.boost.org/doc/libs/1_60_0/libs/math/doc/html/index.html">Boost</a> |
156 | * and the ratio |
157 | * @f[ |
158 | * \frac{|f - f_{test}|}{|f_{test}|} |
159 | * @f] |
160 | * is generally found to be within 10<sup>-15</sup> for 64-bit double on |
161 | * linux-x86_64 systems over most of the ranges of validity. |
162 | * |
163 | * @todo Provide accuracy comparisons on a per-function basis for a small |
164 | * number of targets. |
165 | * |
166 | * @subsection mathsf_bibliography General Bibliography |
167 | * |
168 | * @see Abramowitz and Stegun: Handbook of Mathematical Functions, |
169 | * with Formulas, Graphs, and Mathematical Tables |
170 | * Edited by Milton Abramowitz and Irene A. Stegun, |
171 | * National Bureau of Standards Applied Mathematics Series - 55 |
172 | * Issued June 1964, Tenth Printing, December 1972, with corrections |
173 | * Electronic versions of A&S abound including both pdf and navigable html. |
174 | * @see for example http://people.math.sfu.ca/~cbm/aands/ |
175 | * |
176 | * @see The old A&S has been redone as the |
177 | * NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/ |
178 | * This version is far more navigable and includes more recent work. |
179 | * |
180 | * @see An Atlas of Functions: with Equator, the Atlas Function Calculator |
181 | * 2nd Edition, by Oldham, Keith B., Myland, Jan, Spanier, Jerome |
182 | * |
183 | * @see Asymptotics and Special Functions by Frank W. J. Olver, |
184 | * Academic Press, 1974 |
185 | * |
186 | * @see Numerical Recipes in C, The Art of Scientific Computing, |
187 | * by William H. Press, Second Ed., Saul A. Teukolsky, |
188 | * William T. Vetterling, and Brian P. Flannery, |
189 | * Cambridge University Press, 1992 |
190 | * |
191 | * @see The Special Functions and Their Approximations: Volumes 1 and 2, |
192 | * by Yudell L. Luke, Academic Press, 1969 |
193 | * |
194 | * @{ |
195 | */ |
196 | |
197 | // Associated Laguerre polynomials |
198 | |
199 | /** |
200 | * Return the associated Laguerre polynomial of order @c n, |
201 | * degree @c m: @f$ L_n^m(x) @f$ for @c float argument. |
202 | * |
203 | * @see assoc_laguerre for more details. |
204 | */ |
205 | inline float |
206 | assoc_laguerref(unsigned int __n, unsigned int __m, float __x) |
207 | { return __detail::__assoc_laguerre<float>(__n, __m, __x); } |
208 | |
209 | /** |
210 | * Return the associated Laguerre polynomial of order @c n, |
211 | * degree @c m: @f$ L_n^m(x) @f$. |
212 | * |
213 | * @see assoc_laguerre for more details. |
214 | */ |
215 | inline long double |
216 | assoc_laguerrel(unsigned int __n, unsigned int __m, long double __x) |
217 | { return __detail::__assoc_laguerre<long double>(__n, __m, __x); } |
218 | |
219 | /** |
220 | * Return the associated Laguerre polynomial of nonnegative order @c n, |
221 | * nonnegative degree @c m and real argument @c x: @f$ L_n^m(x) @f$. |
222 | * |
223 | * The associated Laguerre function of real degree @f$ \alpha @f$, |
224 | * @f$ L_n^\alpha(x) @f$, is defined by |
225 | * @f[ |
226 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} |
227 | * {}_1F_1(-n; \alpha + 1; x) |
228 | * @f] |
229 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and |
230 | * @f$ {}_1F_1(a; c; x) @f$ is the confluent hypergeometric function. |
231 | * |
232 | * The associated Laguerre polynomial is defined for integral |
233 | * degree @f$ \alpha = m @f$ by: |
234 | * @f[ |
235 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) |
236 | * @f] |
237 | * where the Laguerre polynomial is defined by: |
238 | * @f[ |
239 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |
240 | * @f] |
241 | * and @f$ x >= 0 @f$. |
242 | * @see laguerre for details of the Laguerre function of degree @c n |
243 | * |
244 | * @tparam _Tp The floating-point type of the argument @c __x. |
245 | * @param __n The order of the Laguerre function, <tt>__n >= 0</tt>. |
246 | * @param __m The degree of the Laguerre function, <tt>__m >= 0</tt>. |
247 | * @param __x The argument of the Laguerre function, <tt>__x >= 0</tt>. |
248 | * @throw std::domain_error if <tt>__x < 0</tt>. |
249 | */ |
250 | template<typename _Tp> |
251 | inline typename __gnu_cxx::__promote<_Tp>::__type |
252 | assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x) |
253 | { |
254 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
255 | return __detail::__assoc_laguerre<__type>(__n, __m, __x); |
256 | } |
257 | |
258 | // Associated Legendre functions |
259 | |
260 | /** |
261 | * Return the associated Legendre function of degree @c l and order @c m |
262 | * for @c float argument. |
263 | * |
264 | * @see assoc_legendre for more details. |
265 | */ |
266 | inline float |
267 | assoc_legendref(unsigned int __l, unsigned int __m, float __x) |
268 | { return __detail::__assoc_legendre_p<float>(__l, __m, __x); } |
269 | |
270 | /** |
271 | * Return the associated Legendre function of degree @c l and order @c m. |
272 | * |
273 | * @see assoc_legendre for more details. |
274 | */ |
275 | inline long double |
276 | assoc_legendrel(unsigned int __l, unsigned int __m, long double __x) |
277 | { return __detail::__assoc_legendre_p<long double>(__l, __m, __x); } |
278 | |
279 | |
280 | /** |
281 | * Return the associated Legendre function of degree @c l and order @c m. |
282 | * |
283 | * The associated Legendre function is derived from the Legendre function |
284 | * @f$ P_l(x) @f$ by the Rodrigues formula: |
285 | * @f[ |
286 | * P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x) |
287 | * @f] |
288 | * @see legendre for details of the Legendre function of degree @c l |
289 | * |
290 | * @tparam _Tp The floating-point type of the argument @c __x. |
291 | * @param __l The degree <tt>__l >= 0</tt>. |
292 | * @param __m The order <tt>__m <= l</tt>. |
293 | * @param __x The argument, <tt>abs(__x) <= 1</tt>. |
294 | * @throw std::domain_error if <tt>abs(__x) > 1</tt>. |
295 | */ |
296 | template<typename _Tp> |
297 | inline typename __gnu_cxx::__promote<_Tp>::__type |
298 | assoc_legendre(unsigned int __l, unsigned int __m, _Tp __x) |
299 | { |
300 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
301 | return __detail::__assoc_legendre_p<__type>(__l, __m, __x); |
302 | } |
303 | |
304 | // Beta functions |
305 | |
306 | /** |
307 | * Return the beta function, @f$ B(a,b) @f$, for @c float parameters @c a, @c b. |
308 | * |
309 | * @see beta for more details. |
310 | */ |
311 | inline float |
312 | betaf(float __a, float __b) |
313 | { return __detail::__beta<float>(x: __a, y: __b); } |
314 | |
315 | /** |
316 | * Return the beta function, @f$B(a,b)@f$, for long double |
317 | * parameters @c a, @c b. |
318 | * |
319 | * @see beta for more details. |
320 | */ |
321 | inline long double |
322 | betal(long double __a, long double __b) |
323 | { return __detail::__beta<long double>(x: __a, y: __b); } |
324 | |
325 | /** |
326 | * Return the beta function, @f$B(a,b)@f$, for real parameters @c a, @c b. |
327 | * |
328 | * The beta function is defined by |
329 | * @f[ |
330 | * B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt |
331 | * = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} |
332 | * @f] |
333 | * where @f$ a > 0 @f$ and @f$ b > 0 @f$ |
334 | * |
335 | * @tparam _Tpa The floating-point type of the parameter @c __a. |
336 | * @tparam _Tpb The floating-point type of the parameter @c __b. |
337 | * @param __a The first argument of the beta function, <tt> __a > 0 </tt>. |
338 | * @param __b The second argument of the beta function, <tt> __b > 0 </tt>. |
339 | * @throw std::domain_error if <tt> __a < 0 </tt> or <tt> __b < 0 </tt>. |
340 | */ |
341 | template<typename _Tpa, typename _Tpb> |
342 | inline typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type |
343 | beta(_Tpa __a, _Tpb __b) |
344 | { |
345 | typedef typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type __type; |
346 | return __detail::__beta<__type>(__a, __b); |
347 | } |
348 | |
349 | // Complete elliptic integrals of the first kind |
350 | |
351 | /** |
352 | * Return the complete elliptic integral of the first kind @f$ E(k) @f$ |
353 | * for @c float modulus @c k. |
354 | * |
355 | * @see comp_ellint_1 for details. |
356 | */ |
357 | inline float |
358 | comp_ellint_1f(float __k) |
359 | { return __detail::__comp_ellint_1<float>(__k); } |
360 | |
361 | /** |
362 | * Return the complete elliptic integral of the first kind @f$ E(k) @f$ |
363 | * for long double modulus @c k. |
364 | * |
365 | * @see comp_ellint_1 for details. |
366 | */ |
367 | inline long double |
368 | comp_ellint_1l(long double __k) |
369 | { return __detail::__comp_ellint_1<long double>(__k); } |
370 | |
371 | /** |
372 | * Return the complete elliptic integral of the first kind |
373 | * @f$ K(k) @f$ for real modulus @c k. |
374 | * |
375 | * The complete elliptic integral of the first kind is defined as |
376 | * @f[ |
377 | * K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} |
378 | * {\sqrt{1 - k^2 sin^2\theta}} |
379 | * @f] |
380 | * where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the |
381 | * first kind and the modulus @f$ |k| <= 1 @f$. |
382 | * @see ellint_1 for details of the incomplete elliptic function |
383 | * of the first kind. |
384 | * |
385 | * @tparam _Tp The floating-point type of the modulus @c __k. |
386 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> |
387 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. |
388 | */ |
389 | template<typename _Tp> |
390 | inline typename __gnu_cxx::__promote<_Tp>::__type |
391 | comp_ellint_1(_Tp __k) |
392 | { |
393 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
394 | return __detail::__comp_ellint_1<__type>(__k); |
395 | } |
396 | |
397 | // Complete elliptic integrals of the second kind |
398 | |
399 | /** |
400 | * Return the complete elliptic integral of the second kind @f$ E(k) @f$ |
401 | * for @c float modulus @c k. |
402 | * |
403 | * @see comp_ellint_2 for details. |
404 | */ |
405 | inline float |
406 | comp_ellint_2f(float __k) |
407 | { return __detail::__comp_ellint_2<float>(__k); } |
408 | |
409 | /** |
410 | * Return the complete elliptic integral of the second kind @f$ E(k) @f$ |
411 | * for long double modulus @c k. |
412 | * |
413 | * @see comp_ellint_2 for details. |
414 | */ |
415 | inline long double |
416 | comp_ellint_2l(long double __k) |
417 | { return __detail::__comp_ellint_2<long double>(__k); } |
418 | |
419 | /** |
420 | * Return the complete elliptic integral of the second kind @f$ E(k) @f$ |
421 | * for real modulus @c k. |
422 | * |
423 | * The complete elliptic integral of the second kind is defined as |
424 | * @f[ |
425 | * E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} |
426 | * @f] |
427 | * where @f$ E(k,\phi) @f$ is the incomplete elliptic integral of the |
428 | * second kind and the modulus @f$ |k| <= 1 @f$. |
429 | * @see ellint_2 for details of the incomplete elliptic function |
430 | * of the second kind. |
431 | * |
432 | * @tparam _Tp The floating-point type of the modulus @c __k. |
433 | * @param __k The modulus, @c abs(__k) <= 1 |
434 | * @throw std::domain_error if @c abs(__k) > 1. |
435 | */ |
436 | template<typename _Tp> |
437 | inline typename __gnu_cxx::__promote<_Tp>::__type |
438 | comp_ellint_2(_Tp __k) |
439 | { |
440 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
441 | return __detail::__comp_ellint_2<__type>(__k); |
442 | } |
443 | |
444 | // Complete elliptic integrals of the third kind |
445 | |
446 | /** |
447 | * @brief Return the complete elliptic integral of the third kind |
448 | * @f$ \Pi(k,\nu) @f$ for @c float modulus @c k. |
449 | * |
450 | * @see comp_ellint_3 for details. |
451 | */ |
452 | inline float |
453 | comp_ellint_3f(float __k, float __nu) |
454 | { return __detail::__comp_ellint_3<float>(__k, __nu); } |
455 | |
456 | /** |
457 | * @brief Return the complete elliptic integral of the third kind |
458 | * @f$ \Pi(k,\nu) @f$ for <tt>long double</tt> modulus @c k. |
459 | * |
460 | * @see comp_ellint_3 for details. |
461 | */ |
462 | inline long double |
463 | comp_ellint_3l(long double __k, long double __nu) |
464 | { return __detail::__comp_ellint_3<long double>(__k, __nu); } |
465 | |
466 | /** |
467 | * Return the complete elliptic integral of the third kind |
468 | * @f$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) @f$ for real modulus @c k. |
469 | * |
470 | * The complete elliptic integral of the third kind is defined as |
471 | * @f[ |
472 | * \Pi(k,\nu) = \Pi(k,\nu,\pi/2) = \int_0^{\pi/2} |
473 | * \frac{d\theta} |
474 | * {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} |
475 | * @f] |
476 | * where @f$ \Pi(k,\nu,\phi) @f$ is the incomplete elliptic integral of the |
477 | * second kind and the modulus @f$ |k| <= 1 @f$. |
478 | * @see ellint_3 for details of the incomplete elliptic function |
479 | * of the third kind. |
480 | * |
481 | * @tparam _Tp The floating-point type of the modulus @c __k. |
482 | * @tparam _Tpn The floating-point type of the argument @c __nu. |
483 | * @param __k The modulus, @c abs(__k) <= 1 |
484 | * @param __nu The argument |
485 | * @throw std::domain_error if @c abs(__k) > 1. |
486 | */ |
487 | template<typename _Tp, typename _Tpn> |
488 | inline typename __gnu_cxx::__promote_2<_Tp, _Tpn>::__type |
489 | comp_ellint_3(_Tp __k, _Tpn __nu) |
490 | { |
491 | typedef typename __gnu_cxx::__promote_2<_Tp, _Tpn>::__type __type; |
492 | return __detail::__comp_ellint_3<__type>(__k, __nu); |
493 | } |
494 | |
495 | // Regular modified cylindrical Bessel functions |
496 | |
497 | /** |
498 | * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ |
499 | * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
500 | * |
501 | * @see cyl_bessel_i for setails. |
502 | */ |
503 | inline float |
504 | cyl_bessel_if(float __nu, float __x) |
505 | { return __detail::__cyl_bessel_i<float>(__nu, __x); } |
506 | |
507 | /** |
508 | * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ |
509 | * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
510 | * |
511 | * @see cyl_bessel_i for setails. |
512 | */ |
513 | inline long double |
514 | cyl_bessel_il(long double __nu, long double __x) |
515 | { return __detail::__cyl_bessel_i<long double>(__nu, __x); } |
516 | |
517 | /** |
518 | * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ |
519 | * for real order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
520 | * |
521 | * The regular modified cylindrical Bessel function is: |
522 | * @f[ |
523 | * I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty} |
524 | * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
525 | * @f] |
526 | * |
527 | * @tparam _Tpnu The floating-point type of the order @c __nu. |
528 | * @tparam _Tp The floating-point type of the argument @c __x. |
529 | * @param __nu The order |
530 | * @param __x The argument, <tt> __x >= 0 </tt> |
531 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
532 | */ |
533 | template<typename _Tpnu, typename _Tp> |
534 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type |
535 | cyl_bessel_i(_Tpnu __nu, _Tp __x) |
536 | { |
537 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; |
538 | return __detail::__cyl_bessel_i<__type>(__nu, __x); |
539 | } |
540 | |
541 | // Cylindrical Bessel functions (of the first kind) |
542 | |
543 | /** |
544 | * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$ |
545 | * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
546 | * |
547 | * @see cyl_bessel_j for setails. |
548 | */ |
549 | inline float |
550 | cyl_bessel_jf(float __nu, float __x) |
551 | { return __detail::__cyl_bessel_j<float>(__nu, __x); } |
552 | |
553 | /** |
554 | * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$ |
555 | * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
556 | * |
557 | * @see cyl_bessel_j for setails. |
558 | */ |
559 | inline long double |
560 | cyl_bessel_jl(long double __nu, long double __x) |
561 | { return __detail::__cyl_bessel_j<long double>(__nu, __x); } |
562 | |
563 | /** |
564 | * Return the Bessel function @f$ J_{\nu}(x) @f$ of real order @f$ \nu @f$ |
565 | * and argument @f$ x >= 0 @f$. |
566 | * |
567 | * The cylindrical Bessel function is: |
568 | * @f[ |
569 | * J_{\nu}(x) = \sum_{k=0}^{\infty} |
570 | * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
571 | * @f] |
572 | * |
573 | * @tparam _Tpnu The floating-point type of the order @c __nu. |
574 | * @tparam _Tp The floating-point type of the argument @c __x. |
575 | * @param __nu The order |
576 | * @param __x The argument, <tt> __x >= 0 </tt> |
577 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
578 | */ |
579 | template<typename _Tpnu, typename _Tp> |
580 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type |
581 | cyl_bessel_j(_Tpnu __nu, _Tp __x) |
582 | { |
583 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; |
584 | return __detail::__cyl_bessel_j<__type>(__nu, __x); |
585 | } |
586 | |
587 | // Irregular modified cylindrical Bessel functions |
588 | |
589 | /** |
590 | * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ |
591 | * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
592 | * |
593 | * @see cyl_bessel_k for setails. |
594 | */ |
595 | inline float |
596 | cyl_bessel_kf(float __nu, float __x) |
597 | { return __detail::__cyl_bessel_k<float>(__nu, __x); } |
598 | |
599 | /** |
600 | * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ |
601 | * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
602 | * |
603 | * @see cyl_bessel_k for setails. |
604 | */ |
605 | inline long double |
606 | cyl_bessel_kl(long double __nu, long double __x) |
607 | { return __detail::__cyl_bessel_k<long double>(__nu, __x); } |
608 | |
609 | /** |
610 | * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ |
611 | * of real order @f$ \nu @f$ and argument @f$ x @f$. |
612 | * |
613 | * The irregular modified Bessel function is defined by: |
614 | * @f[ |
615 | * K_{\nu}(x) = \frac{\pi}{2} |
616 | * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} |
617 | * @f] |
618 | * where for integral @f$ \nu = n @f$ a limit is taken: |
619 | * @f$ lim_{\nu \to n} @f$. |
620 | * For negative argument we have simply: |
621 | * @f[ |
622 | * K_{-\nu}(x) = K_{\nu}(x) |
623 | * @f] |
624 | * |
625 | * @tparam _Tpnu The floating-point type of the order @c __nu. |
626 | * @tparam _Tp The floating-point type of the argument @c __x. |
627 | * @param __nu The order |
628 | * @param __x The argument, <tt> __x >= 0 </tt> |
629 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
630 | */ |
631 | template<typename _Tpnu, typename _Tp> |
632 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type |
633 | cyl_bessel_k(_Tpnu __nu, _Tp __x) |
634 | { |
635 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; |
636 | return __detail::__cyl_bessel_k<__type>(__nu, __x); |
637 | } |
638 | |
639 | // Cylindrical Neumann functions |
640 | |
641 | /** |
642 | * Return the Neumann function @f$ N_{\nu}(x) @f$ |
643 | * of @c float order @f$ \nu @f$ and argument @f$ x @f$. |
644 | * |
645 | * @see cyl_neumann for setails. |
646 | */ |
647 | inline float |
648 | cyl_neumannf(float __nu, float __x) |
649 | { return __detail::__cyl_neumann_n<float>(__nu, __x); } |
650 | |
651 | /** |
652 | * Return the Neumann function @f$ N_{\nu}(x) @f$ |
653 | * of <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x @f$. |
654 | * |
655 | * @see cyl_neumann for setails. |
656 | */ |
657 | inline long double |
658 | cyl_neumannl(long double __nu, long double __x) |
659 | { return __detail::__cyl_neumann_n<long double>(__nu, __x); } |
660 | |
661 | /** |
662 | * Return the Neumann function @f$ N_{\nu}(x) @f$ |
663 | * of real order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
664 | * |
665 | * The Neumann function is defined by: |
666 | * @f[ |
667 | * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} |
668 | * {\sin \nu\pi} |
669 | * @f] |
670 | * where @f$ x >= 0 @f$ and for integral order @f$ \nu = n @f$ |
671 | * a limit is taken: @f$ lim_{\nu \to n} @f$. |
672 | * |
673 | * @tparam _Tpnu The floating-point type of the order @c __nu. |
674 | * @tparam _Tp The floating-point type of the argument @c __x. |
675 | * @param __nu The order |
676 | * @param __x The argument, <tt> __x >= 0 </tt> |
677 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
678 | */ |
679 | template<typename _Tpnu, typename _Tp> |
680 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type |
681 | cyl_neumann(_Tpnu __nu, _Tp __x) |
682 | { |
683 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; |
684 | return __detail::__cyl_neumann_n<__type>(__nu, __x); |
685 | } |
686 | |
687 | // Incomplete elliptic integrals of the first kind |
688 | |
689 | /** |
690 | * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$ |
691 | * for @c float modulus @f$ k @f$ and angle @f$ \phi @f$. |
692 | * |
693 | * @see ellint_1 for details. |
694 | */ |
695 | inline float |
696 | ellint_1f(float __k, float __phi) |
697 | { return __detail::__ellint_1<float>(__k, __phi); } |
698 | |
699 | /** |
700 | * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$ |
701 | * for <tt>long double</tt> modulus @f$ k @f$ and angle @f$ \phi @f$. |
702 | * |
703 | * @see ellint_1 for details. |
704 | */ |
705 | inline long double |
706 | ellint_1l(long double __k, long double __phi) |
707 | { return __detail::__ellint_1<long double>(__k, __phi); } |
708 | |
709 | /** |
710 | * Return the incomplete elliptic integral of the first kind @f$ F(k,\phi) @f$ |
711 | * for @c real modulus @f$ k @f$ and angle @f$ \phi @f$. |
712 | * |
713 | * The incomplete elliptic integral of the first kind is defined as |
714 | * @f[ |
715 | * F(k,\phi) = \int_0^{\phi}\frac{d\theta} |
716 | * {\sqrt{1 - k^2 sin^2\theta}} |
717 | * @f] |
718 | * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of |
719 | * the first kind, @f$ K(k) @f$. @see comp_ellint_1. |
720 | * |
721 | * @tparam _Tp The floating-point type of the modulus @c __k. |
722 | * @tparam _Tpp The floating-point type of the angle @c __phi. |
723 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> |
724 | * @param __phi The integral limit argument in radians |
725 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. |
726 | */ |
727 | template<typename _Tp, typename _Tpp> |
728 | inline typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type |
729 | ellint_1(_Tp __k, _Tpp __phi) |
730 | { |
731 | typedef typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type __type; |
732 | return __detail::__ellint_1<__type>(__k, __phi); |
733 | } |
734 | |
735 | // Incomplete elliptic integrals of the second kind |
736 | |
737 | /** |
738 | * @brief Return the incomplete elliptic integral of the second kind |
739 | * @f$ E(k,\phi) @f$ for @c float argument. |
740 | * |
741 | * @see ellint_2 for details. |
742 | */ |
743 | inline float |
744 | ellint_2f(float __k, float __phi) |
745 | { return __detail::__ellint_2<float>(__k, __phi); } |
746 | |
747 | /** |
748 | * @brief Return the incomplete elliptic integral of the second kind |
749 | * @f$ E(k,\phi) @f$. |
750 | * |
751 | * @see ellint_2 for details. |
752 | */ |
753 | inline long double |
754 | ellint_2l(long double __k, long double __phi) |
755 | { return __detail::__ellint_2<long double>(__k, __phi); } |
756 | |
757 | /** |
758 | * Return the incomplete elliptic integral of the second kind |
759 | * @f$ E(k,\phi) @f$. |
760 | * |
761 | * The incomplete elliptic integral of the second kind is defined as |
762 | * @f[ |
763 | * E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} |
764 | * @f] |
765 | * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of |
766 | * the second kind, @f$ E(k) @f$. @see comp_ellint_2. |
767 | * |
768 | * @tparam _Tp The floating-point type of the modulus @c __k. |
769 | * @tparam _Tpp The floating-point type of the angle @c __phi. |
770 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> |
771 | * @param __phi The integral limit argument in radians |
772 | * @return The elliptic function of the second kind. |
773 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. |
774 | */ |
775 | template<typename _Tp, typename _Tpp> |
776 | inline typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type |
777 | ellint_2(_Tp __k, _Tpp __phi) |
778 | { |
779 | typedef typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type __type; |
780 | return __detail::__ellint_2<__type>(__k, __phi); |
781 | } |
782 | |
783 | // Incomplete elliptic integrals of the third kind |
784 | |
785 | /** |
786 | * @brief Return the incomplete elliptic integral of the third kind |
787 | * @f$ \Pi(k,\nu,\phi) @f$ for @c float argument. |
788 | * |
789 | * @see ellint_3 for details. |
790 | */ |
791 | inline float |
792 | ellint_3f(float __k, float __nu, float __phi) |
793 | { return __detail::__ellint_3<float>(__k, __nu, __phi); } |
794 | |
795 | /** |
796 | * @brief Return the incomplete elliptic integral of the third kind |
797 | * @f$ \Pi(k,\nu,\phi) @f$. |
798 | * |
799 | * @see ellint_3 for details. |
800 | */ |
801 | inline long double |
802 | ellint_3l(long double __k, long double __nu, long double __phi) |
803 | { return __detail::__ellint_3<long double>(__k, __nu, __phi); } |
804 | |
805 | /** |
806 | * @brief Return the incomplete elliptic integral of the third kind |
807 | * @f$ \Pi(k,\nu,\phi) @f$. |
808 | * |
809 | * The incomplete elliptic integral of the third kind is defined by: |
810 | * @f[ |
811 | * \Pi(k,\nu,\phi) = \int_0^{\phi} |
812 | * \frac{d\theta} |
813 | * {(1 - \nu \sin^2\theta) |
814 | * \sqrt{1 - k^2 \sin^2\theta}} |
815 | * @f] |
816 | * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of |
817 | * the third kind, @f$ \Pi(k,\nu) @f$. @see comp_ellint_3. |
818 | * |
819 | * @tparam _Tp The floating-point type of the modulus @c __k. |
820 | * @tparam _Tpn The floating-point type of the argument @c __nu. |
821 | * @tparam _Tpp The floating-point type of the angle @c __phi. |
822 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> |
823 | * @param __nu The second argument |
824 | * @param __phi The integral limit argument in radians |
825 | * @return The elliptic function of the third kind. |
826 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. |
827 | */ |
828 | template<typename _Tp, typename _Tpn, typename _Tpp> |
829 | inline typename __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type |
830 | ellint_3(_Tp __k, _Tpn __nu, _Tpp __phi) |
831 | { |
832 | typedef typename __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type __type; |
833 | return __detail::__ellint_3<__type>(__k, __nu, __phi); |
834 | } |
835 | |
836 | // Exponential integrals |
837 | |
838 | /** |
839 | * Return the exponential integral @f$ Ei(x) @f$ for @c float argument @c x. |
840 | * |
841 | * @see expint for details. |
842 | */ |
843 | inline float |
844 | expintf(float __x) |
845 | { return __detail::__expint<float>(__x); } |
846 | |
847 | /** |
848 | * Return the exponential integral @f$ Ei(x) @f$ |
849 | * for <tt>long double</tt> argument @c x. |
850 | * |
851 | * @see expint for details. |
852 | */ |
853 | inline long double |
854 | expintl(long double __x) |
855 | { return __detail::__expint<long double>(__x); } |
856 | |
857 | /** |
858 | * Return the exponential integral @f$ Ei(x) @f$ for @c real argument @c x. |
859 | * |
860 | * The exponential integral is given by |
861 | * \f[ |
862 | * Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt |
863 | * \f] |
864 | * |
865 | * @tparam _Tp The floating-point type of the argument @c __x. |
866 | * @param __x The argument of the exponential integral function. |
867 | */ |
868 | template<typename _Tp> |
869 | inline typename __gnu_cxx::__promote<_Tp>::__type |
870 | expint(_Tp __x) |
871 | { |
872 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
873 | return __detail::__expint<__type>(__x); |
874 | } |
875 | |
876 | // Hermite polynomials |
877 | |
878 | /** |
879 | * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n |
880 | * and float argument @c x. |
881 | * |
882 | * @see hermite for details. |
883 | */ |
884 | inline float |
885 | hermitef(unsigned int __n, float __x) |
886 | { return __detail::__poly_hermite<float>(__n, __x); } |
887 | |
888 | /** |
889 | * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n |
890 | * and <tt>long double</tt> argument @c x. |
891 | * |
892 | * @see hermite for details. |
893 | */ |
894 | inline long double |
895 | hermitel(unsigned int __n, long double __x) |
896 | { return __detail::__poly_hermite<long double>(__n, __x); } |
897 | |
898 | /** |
899 | * Return the Hermite polynomial @f$ H_n(x) @f$ of order n |
900 | * and @c real argument @c x. |
901 | * |
902 | * The Hermite polynomial is defined by: |
903 | * @f[ |
904 | * H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} |
905 | * @f] |
906 | * |
907 | * The Hermite polynomial obeys a reflection formula: |
908 | * @f[ |
909 | * H_n(-x) = (-1)^n H_n(x) |
910 | * @f] |
911 | * |
912 | * @tparam _Tp The floating-point type of the argument @c __x. |
913 | * @param __n The order |
914 | * @param __x The argument |
915 | */ |
916 | template<typename _Tp> |
917 | inline typename __gnu_cxx::__promote<_Tp>::__type |
918 | hermite(unsigned int __n, _Tp __x) |
919 | { |
920 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
921 | return __detail::__poly_hermite<__type>(__n, __x); |
922 | } |
923 | |
924 | // Laguerre polynomials |
925 | |
926 | /** |
927 | * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n |
928 | * and @c float argument @f$ x >= 0 @f$. |
929 | * |
930 | * @see laguerre for more details. |
931 | */ |
932 | inline float |
933 | laguerref(unsigned int __n, float __x) |
934 | { return __detail::__laguerre<float>(__n, __x); } |
935 | |
936 | /** |
937 | * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n |
938 | * and <tt>long double</tt> argument @f$ x >= 0 @f$. |
939 | * |
940 | * @see laguerre for more details. |
941 | */ |
942 | inline long double |
943 | laguerrel(unsigned int __n, long double __x) |
944 | { return __detail::__laguerre<long double>(__n, __x); } |
945 | |
946 | /** |
947 | * Returns the Laguerre polynomial @f$ L_n(x) @f$ |
948 | * of nonnegative degree @c n and real argument @f$ x >= 0 @f$. |
949 | * |
950 | * The Laguerre polynomial is defined by: |
951 | * @f[ |
952 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |
953 | * @f] |
954 | * |
955 | * @tparam _Tp The floating-point type of the argument @c __x. |
956 | * @param __n The nonnegative order |
957 | * @param __x The argument <tt> __x >= 0 </tt> |
958 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
959 | */ |
960 | template<typename _Tp> |
961 | inline typename __gnu_cxx::__promote<_Tp>::__type |
962 | laguerre(unsigned int __n, _Tp __x) |
963 | { |
964 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
965 | return __detail::__laguerre<__type>(__n, __x); |
966 | } |
967 | |
968 | // Legendre polynomials |
969 | |
970 | /** |
971 | * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative |
972 | * degree @f$ l @f$ and @c float argument @f$ |x| <= 0 @f$. |
973 | * |
974 | * @see legendre for more details. |
975 | */ |
976 | inline float |
977 | legendref(unsigned int __l, float __x) |
978 | { return __detail::__poly_legendre_p<float>(__l, __x); } |
979 | |
980 | /** |
981 | * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative |
982 | * degree @f$ l @f$ and <tt>long double</tt> argument @f$ |x| <= 0 @f$. |
983 | * |
984 | * @see legendre for more details. |
985 | */ |
986 | inline long double |
987 | legendrel(unsigned int __l, long double __x) |
988 | { return __detail::__poly_legendre_p<long double>(__l, __x); } |
989 | |
990 | /** |
991 | * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative |
992 | * degree @f$ l @f$ and real argument @f$ |x| <= 0 @f$. |
993 | * |
994 | * The Legendre function of order @f$ l @f$ and argument @f$ x @f$, |
995 | * @f$ P_l(x) @f$, is defined by: |
996 | * @f[ |
997 | * P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l} |
998 | * @f] |
999 | * |
1000 | * @tparam _Tp The floating-point type of the argument @c __x. |
1001 | * @param __l The degree @f$ l >= 0 @f$ |
1002 | * @param __x The argument @c abs(__x) <= 1 |
1003 | * @throw std::domain_error if @c abs(__x) > 1 |
1004 | */ |
1005 | template<typename _Tp> |
1006 | inline typename __gnu_cxx::__promote<_Tp>::__type |
1007 | legendre(unsigned int __l, _Tp __x) |
1008 | { |
1009 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
1010 | return __detail::__poly_legendre_p<__type>(__l, __x); |
1011 | } |
1012 | |
1013 | // Riemann zeta functions |
1014 | |
1015 | /** |
1016 | * Return the Riemann zeta function @f$ \zeta(s) @f$ |
1017 | * for @c float argument @f$ s @f$. |
1018 | * |
1019 | * @see riemann_zeta for more details. |
1020 | */ |
1021 | inline float |
1022 | riemann_zetaf(float __s) |
1023 | { return __detail::__riemann_zeta<float>(__s); } |
1024 | |
1025 | /** |
1026 | * Return the Riemann zeta function @f$ \zeta(s) @f$ |
1027 | * for <tt>long double</tt> argument @f$ s @f$. |
1028 | * |
1029 | * @see riemann_zeta for more details. |
1030 | */ |
1031 | inline long double |
1032 | riemann_zetal(long double __s) |
1033 | { return __detail::__riemann_zeta<long double>(__s); } |
1034 | |
1035 | /** |
1036 | * Return the Riemann zeta function @f$ \zeta(s) @f$ |
1037 | * for real argument @f$ s @f$. |
1038 | * |
1039 | * The Riemann zeta function is defined by: |
1040 | * @f[ |
1041 | * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1 |
1042 | * @f] |
1043 | * and |
1044 | * @f[ |
1045 | * \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s} |
1046 | * \hbox{ for } 0 <= s <= 1 |
1047 | * @f] |
1048 | * For s < 1 use the reflection formula: |
1049 | * @f[ |
1050 | * \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s) |
1051 | * @f] |
1052 | * |
1053 | * @tparam _Tp The floating-point type of the argument @c __s. |
1054 | * @param __s The argument <tt> s != 1 </tt> |
1055 | */ |
1056 | template<typename _Tp> |
1057 | inline typename __gnu_cxx::__promote<_Tp>::__type |
1058 | riemann_zeta(_Tp __s) |
1059 | { |
1060 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
1061 | return __detail::__riemann_zeta<__type>(__s); |
1062 | } |
1063 | |
1064 | // Spherical Bessel functions |
1065 | |
1066 | /** |
1067 | * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n |
1068 | * and @c float argument @f$ x >= 0 @f$. |
1069 | * |
1070 | * @see sph_bessel for more details. |
1071 | */ |
1072 | inline float |
1073 | sph_besself(unsigned int __n, float __x) |
1074 | { return __detail::__sph_bessel<float>(__n, __x); } |
1075 | |
1076 | /** |
1077 | * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n |
1078 | * and <tt>long double</tt> argument @f$ x >= 0 @f$. |
1079 | * |
1080 | * @see sph_bessel for more details. |
1081 | */ |
1082 | inline long double |
1083 | sph_bessell(unsigned int __n, long double __x) |
1084 | { return __detail::__sph_bessel<long double>(__n, __x); } |
1085 | |
1086 | /** |
1087 | * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n |
1088 | * and real argument @f$ x >= 0 @f$. |
1089 | * |
1090 | * The spherical Bessel function is defined by: |
1091 | * @f[ |
1092 | * j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) |
1093 | * @f] |
1094 | * |
1095 | * @tparam _Tp The floating-point type of the argument @c __x. |
1096 | * @param __n The integral order <tt> n >= 0 </tt> |
1097 | * @param __x The real argument <tt> x >= 0 </tt> |
1098 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
1099 | */ |
1100 | template<typename _Tp> |
1101 | inline typename __gnu_cxx::__promote<_Tp>::__type |
1102 | sph_bessel(unsigned int __n, _Tp __x) |
1103 | { |
1104 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
1105 | return __detail::__sph_bessel<__type>(__n, __x); |
1106 | } |
1107 | |
1108 | // Spherical associated Legendre functions |
1109 | |
1110 | /** |
1111 | * Return the spherical Legendre function of nonnegative integral |
1112 | * degree @c l and order @c m and float angle @f$ \theta @f$ in radians. |
1113 | * |
1114 | * @see sph_legendre for details. |
1115 | */ |
1116 | inline float |
1117 | sph_legendref(unsigned int __l, unsigned int __m, float __theta) |
1118 | { return __detail::__sph_legendre<float>(__l, __m, __theta); } |
1119 | |
1120 | /** |
1121 | * Return the spherical Legendre function of nonnegative integral |
1122 | * degree @c l and order @c m and <tt>long double</tt> angle @f$ \theta @f$ |
1123 | * in radians. |
1124 | * |
1125 | * @see sph_legendre for details. |
1126 | */ |
1127 | inline long double |
1128 | sph_legendrel(unsigned int __l, unsigned int __m, long double __theta) |
1129 | { return __detail::__sph_legendre<long double>(__l, __m, __theta); } |
1130 | |
1131 | /** |
1132 | * Return the spherical Legendre function of nonnegative integral |
1133 | * degree @c l and order @c m and real angle @f$ \theta @f$ in radians. |
1134 | * |
1135 | * The spherical Legendre function is defined by |
1136 | * @f[ |
1137 | * Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi} |
1138 | * \frac{(l-m)!}{(l+m)!}] |
1139 | * P_l^m(\cos\theta) \exp^{im\phi} |
1140 | * @f] |
1141 | * |
1142 | * @tparam _Tp The floating-point type of the angle @c __theta. |
1143 | * @param __l The order <tt> __l >= 0 </tt> |
1144 | * @param __m The degree <tt> __m >= 0 </tt> and <tt> __m <= __l </tt> |
1145 | * @param __theta The radian polar angle argument |
1146 | */ |
1147 | template<typename _Tp> |
1148 | inline typename __gnu_cxx::__promote<_Tp>::__type |
1149 | sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta) |
1150 | { |
1151 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
1152 | return __detail::__sph_legendre<__type>(__l, __m, __theta); |
1153 | } |
1154 | |
1155 | // Spherical Neumann functions |
1156 | |
1157 | /** |
1158 | * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ |
1159 | * and @c float argument @f$ x >= 0 @f$. |
1160 | * |
1161 | * @see sph_neumann for details. |
1162 | */ |
1163 | inline float |
1164 | sph_neumannf(unsigned int __n, float __x) |
1165 | { return __detail::__sph_neumann<float>(__n, __x); } |
1166 | |
1167 | /** |
1168 | * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ |
1169 | * and <tt>long double</tt> @f$ x >= 0 @f$. |
1170 | * |
1171 | * @see sph_neumann for details. |
1172 | */ |
1173 | inline long double |
1174 | sph_neumannl(unsigned int __n, long double __x) |
1175 | { return __detail::__sph_neumann<long double>(__n, __x); } |
1176 | |
1177 | /** |
1178 | * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ |
1179 | * and real argument @f$ x >= 0 @f$. |
1180 | * |
1181 | * The spherical Neumann function is defined by |
1182 | * @f[ |
1183 | * n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) |
1184 | * @f] |
1185 | * |
1186 | * @tparam _Tp The floating-point type of the argument @c __x. |
1187 | * @param __n The integral order <tt> n >= 0 </tt> |
1188 | * @param __x The real argument <tt> __x >= 0 </tt> |
1189 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
1190 | */ |
1191 | template<typename _Tp> |
1192 | inline typename __gnu_cxx::__promote<_Tp>::__type |
1193 | sph_neumann(unsigned int __n, _Tp __x) |
1194 | { |
1195 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
1196 | return __detail::__sph_neumann<__type>(__n, __x); |
1197 | } |
1198 | |
1199 | /// @} group mathsf |
1200 | |
1201 | _GLIBCXX_END_NAMESPACE_VERSION |
1202 | } // namespace std |
1203 | |
1204 | #ifndef __STRICT_ANSI__ |
1205 | namespace __gnu_cxx _GLIBCXX_VISIBILITY(default) |
1206 | { |
1207 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
1208 | |
1209 | /** @addtogroup mathsf |
1210 | * @{ |
1211 | */ |
1212 | |
1213 | // Airy functions |
1214 | |
1215 | /** |
1216 | * Return the Airy function @f$ Ai(x) @f$ of @c float argument x. |
1217 | */ |
1218 | inline float |
1219 | airy_aif(float __x) |
1220 | { |
1221 | float __Ai, __Bi, __Aip, __Bip; |
1222 | std::__detail::__airy<float>(__x, __Ai, __Bi, __Aip, __Bip); |
1223 | return __Ai; |
1224 | } |
1225 | |
1226 | /** |
1227 | * Return the Airy function @f$ Ai(x) @f$ of <tt>long double</tt> argument x. |
1228 | */ |
1229 | inline long double |
1230 | airy_ail(long double __x) |
1231 | { |
1232 | long double __Ai, __Bi, __Aip, __Bip; |
1233 | std::__detail::__airy<long double>(__x, __Ai, __Bi, __Aip, __Bip); |
1234 | return __Ai; |
1235 | } |
1236 | |
1237 | /** |
1238 | * Return the Airy function @f$ Ai(x) @f$ of real argument x. |
1239 | */ |
1240 | template<typename _Tp> |
1241 | inline typename __gnu_cxx::__promote<_Tp>::__type |
1242 | airy_ai(_Tp __x) |
1243 | { |
1244 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
1245 | __type __Ai, __Bi, __Aip, __Bip; |
1246 | std::__detail::__airy<__type>(__x, __Ai, __Bi, __Aip, __Bip); |
1247 | return __Ai; |
1248 | } |
1249 | |
1250 | /** |
1251 | * Return the Airy function @f$ Bi(x) @f$ of @c float argument x. |
1252 | */ |
1253 | inline float |
1254 | airy_bif(float __x) |
1255 | { |
1256 | float __Ai, __Bi, __Aip, __Bip; |
1257 | std::__detail::__airy<float>(__x, __Ai, __Bi, __Aip, __Bip); |
1258 | return __Bi; |
1259 | } |
1260 | |
1261 | /** |
1262 | * Return the Airy function @f$ Bi(x) @f$ of <tt>long double</tt> argument x. |
1263 | */ |
1264 | inline long double |
1265 | airy_bil(long double __x) |
1266 | { |
1267 | long double __Ai, __Bi, __Aip, __Bip; |
1268 | std::__detail::__airy<long double>(__x, __Ai, __Bi, __Aip, __Bip); |
1269 | return __Bi; |
1270 | } |
1271 | |
1272 | /** |
1273 | * Return the Airy function @f$ Bi(x) @f$ of real argument x. |
1274 | */ |
1275 | template<typename _Tp> |
1276 | inline typename __gnu_cxx::__promote<_Tp>::__type |
1277 | airy_bi(_Tp __x) |
1278 | { |
1279 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
1280 | __type __Ai, __Bi, __Aip, __Bip; |
1281 | std::__detail::__airy<__type>(__x, __Ai, __Bi, __Aip, __Bip); |
1282 | return __Bi; |
1283 | } |
1284 | |
1285 | // Confluent hypergeometric functions |
1286 | |
1287 | /** |
1288 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ |
1289 | * of @c float numeratorial parameter @c a, denominatorial parameter @c c, |
1290 | * and argument @c x. |
1291 | * |
1292 | * @see conf_hyperg for details. |
1293 | */ |
1294 | inline float |
1295 | conf_hypergf(float __a, float __c, float __x) |
1296 | { return std::__detail::__conf_hyperg<float>(__a, __c, __x); } |
1297 | |
1298 | /** |
1299 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ |
1300 | * of <tt>long double</tt> numeratorial parameter @c a, |
1301 | * denominatorial parameter @c c, and argument @c x. |
1302 | * |
1303 | * @see conf_hyperg for details. |
1304 | */ |
1305 | inline long double |
1306 | conf_hypergl(long double __a, long double __c, long double __x) |
1307 | { return std::__detail::__conf_hyperg<long double>(__a, __c, __x); } |
1308 | |
1309 | /** |
1310 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ |
1311 | * of real numeratorial parameter @c a, denominatorial parameter @c c, |
1312 | * and argument @c x. |
1313 | * |
1314 | * The confluent hypergeometric function is defined by |
1315 | * @f[ |
1316 | * {}_1F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n x^n}{(c)_n n!} |
1317 | * @f] |
1318 | * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$, |
1319 | * @f$ (x)_0 = 1 @f$ |
1320 | * |
1321 | * @param __a The numeratorial parameter |
1322 | * @param __c The denominatorial parameter |
1323 | * @param __x The argument |
1324 | */ |
1325 | template<typename _Tpa, typename _Tpc, typename _Tp> |
1326 | inline typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type |
1327 | conf_hyperg(_Tpa __a, _Tpc __c, _Tp __x) |
1328 | { |
1329 | typedef typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type __type; |
1330 | return std::__detail::__conf_hyperg<__type>(__a, __c, __x); |
1331 | } |
1332 | |
1333 | // Hypergeometric functions |
1334 | |
1335 | /** |
1336 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ |
1337 | * of @ float numeratorial parameters @c a and @c b, |
1338 | * denominatorial parameter @c c, and argument @c x. |
1339 | * |
1340 | * @see hyperg for details. |
1341 | */ |
1342 | inline float |
1343 | hypergf(float __a, float __b, float __c, float __x) |
1344 | { return std::__detail::__hyperg<float>(__a, __b, __c, __x); } |
1345 | |
1346 | /** |
1347 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ |
1348 | * of <tt>long double</tt> numeratorial parameters @c a and @c b, |
1349 | * denominatorial parameter @c c, and argument @c x. |
1350 | * |
1351 | * @see hyperg for details. |
1352 | */ |
1353 | inline long double |
1354 | hypergl(long double __a, long double __b, long double __c, long double __x) |
1355 | { return std::__detail::__hyperg<long double>(__a, __b, __c, __x); } |
1356 | |
1357 | /** |
1358 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ |
1359 | * of real numeratorial parameters @c a and @c b, |
1360 | * denominatorial parameter @c c, and argument @c x. |
1361 | * |
1362 | * The hypergeometric function is defined by |
1363 | * @f[ |
1364 | * {}_2F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n x^n}{(c)_n n!} |
1365 | * @f] |
1366 | * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$, |
1367 | * @f$ (x)_0 = 1 @f$ |
1368 | * |
1369 | * @param __a The first numeratorial parameter |
1370 | * @param __b The second numeratorial parameter |
1371 | * @param __c The denominatorial parameter |
1372 | * @param __x The argument |
1373 | */ |
1374 | template<typename _Tpa, typename _Tpb, typename _Tpc, typename _Tp> |
1375 | inline typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp>::__type |
1376 | hyperg(_Tpa __a, _Tpb __b, _Tpc __c, _Tp __x) |
1377 | { |
1378 | typedef typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp> |
1379 | ::__type __type; |
1380 | return std::__detail::__hyperg<__type>(__a, __b, __c, __x); |
1381 | } |
1382 | |
1383 | /// @} |
1384 | _GLIBCXX_END_NAMESPACE_VERSION |
1385 | } // namespace __gnu_cxx |
1386 | #endif // __STRICT_ANSI__ |
1387 | |
1388 | #pragma GCC visibility pop |
1389 | |
1390 | #endif // _GLIBCXX_BITS_SPECFUN_H |
1391 | |