1 | // Mathematical Special Functions for -*- C++ -*-  |
2 |   |
3 | // Copyright (C) 2006-2019 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 | * A collection of advanced mathematical special functions,  |
70 | * defined by ISO/IEC IS 29124.  |
71 | * @{  |
72 | */  |
73 |   |
74 | /**  |
75 | * @mainpage Mathematical Special Functions  |
76 | *  |
77 | * @section intro Introduction and History  |
78 | * The first significant library upgrade on the road to C++2011,  |
79 | * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2005/n1836.pdf">  |
80 | * TR1</a>, included a set of 23 mathematical functions that significantly  |
81 | * extended the standard transcendental functions inherited from C and declared  |
82 | * in @<cmath@>.  |
83 | *  |
84 | * Although most components from TR1 were eventually adopted for C++11 these  |
85 | * math functions were left behind out of concern for implementability.  |
86 | * The math functions were published as a separate international standard  |
87 | * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2010/n3060.pdf">  |
88 | * IS 29124 - Extensions to the C++ Library to Support Mathematical Special  |
89 | * Functions</a>.  |
90 | *  |
91 | * For C++17 these functions were incorporated into the main standard.  |
92 | *  |
93 | * @section contents Contents  |
94 | * The following functions are implemented in namespace @c std:  |
95 | * - @ref assoc_laguerre "assoc_laguerre - Associated Laguerre functions"  |
96 | * - @ref assoc_legendre "assoc_legendre - Associated Legendre functions"  |
97 | * - @ref beta "beta - Beta functions"  |
98 | * - @ref comp_ellint_1 "comp_ellint_1 - Complete elliptic functions of the first kind"  |
99 | * - @ref comp_ellint_2 "comp_ellint_2 - Complete elliptic functions of the second kind"  |
100 | * - @ref comp_ellint_3 "comp_ellint_3 - Complete elliptic functions of the third kind"  |
101 | * - @ref cyl_bessel_i "cyl_bessel_i - Regular modified cylindrical Bessel functions"  |
102 | * - @ref cyl_bessel_j "cyl_bessel_j - Cylindrical Bessel functions of the first kind"  |
103 | * - @ref cyl_bessel_k "cyl_bessel_k - Irregular modified cylindrical Bessel functions"  |
104 | * - @ref cyl_neumann "cyl_neumann - Cylindrical Neumann functions or Cylindrical Bessel functions of the second kind"  |
105 | * - @ref ellint_1 "ellint_1 - Incomplete elliptic functions of the first kind"  |
106 | * - @ref ellint_2 "ellint_2 - Incomplete elliptic functions of the second kind"  |
107 | * - @ref ellint_3 "ellint_3 - Incomplete elliptic functions of the third kind"  |
108 | * - @ref expint "expint - The exponential integral"  |
109 | * - @ref hermite "hermite - Hermite polynomials"  |
110 | * - @ref laguerre "laguerre - Laguerre functions"  |
111 | * - @ref legendre "legendre - Legendre polynomials"  |
112 | * - @ref riemann_zeta "riemann_zeta - The Riemann zeta function"  |
113 | * - @ref sph_bessel "sph_bessel - Spherical Bessel functions"  |
114 | * - @ref sph_legendre "sph_legendre - Spherical Legendre functions"  |
115 | * - @ref sph_neumann "sph_neumann - Spherical Neumann functions"  |
116 | *  |
117 | * The hypergeometric functions were stricken from the TR29124 and C++17  |
118 | * versions of this math library because of implementation concerns.  |
119 | * However, since they were in the TR1 version and since they are popular  |
120 | * we kept them as an extension in namespace @c __gnu_cxx:  |
121 | * - @ref __gnu_cxx::conf_hyperg "conf_hyperg - Confluent hypergeometric functions"  |
122 | * - @ref __gnu_cxx::hyperg "hyperg - Hypergeometric functions"  |
123 | *  |
124 | * @section general General Features  |
125 | *  |
126 | * @subsection promotion Argument Promotion  |
127 | * The arguments suppled to the non-suffixed functions will be promoted  |
128 | * according to the following rules:  |
129 | * 1. If any argument intended to be floating point is given an integral value  |
130 | * That integral value is promoted to double.  |
131 | * 2. All floating point arguments are promoted up to the largest floating  |
132 | * point precision among them.  |
133 | *  |
134 | * @subsection NaN NaN Arguments  |
135 | * If any of the floating point arguments supplied to these functions is  |
136 | * invalid or NaN (std::numeric_limits<Tp>::quiet_NaN),  |
137 | * the value NaN is returned.  |
138 | *  |
139 | * @section impl Implementation  |
140 | *  |
141 | * We strive to implement the underlying math with type generic algorithms  |
142 | * to the greatest extent possible. In practice, the functions are thin  |
143 | * wrappers that dispatch to function templates. Type dependence is  |
144 | * controlled with std::numeric_limits and functions thereof.  |
145 | *  |
146 | * We don't promote @c float to @c double or @c double to <tt>long double</tt>  |
147 | * reflexively. The goal is for @c float functions to operate more quickly,  |
148 | * at the cost of @c float accuracy and possibly a smaller domain of validity.  |
149 | * Similaryly, <tt>long double</tt> should give you more dynamic range  |
150 | * and slightly more pecision than @c double on many systems.  |
151 | *  |
152 | * @section testing Testing  |
153 | *  |
154 | * These functions have been tested against equivalent implementations  |
155 | * from the <a href="http://www.gnu.org/software/gsl">  |
156 | * Gnu Scientific Library, GSL</a> and  |
157 | * <a href="http://www.boost.org/doc/libs/1_60_0/libs/math/doc/html/index.html>Boost</a>  |
158 | * and the ratio  |
159 | * @f[  |
160 | * \frac{|f - f_{test}|}{|f_{test}|}  |
161 | * @f]  |
162 | * is generally found to be within 10^-15 for 64-bit double on linux-x86_64 systems  |
163 | * over most of the ranges of validity.  |
164 | *   |
165 | * @todo Provide accuracy comparisons on a per-function basis for a small  |
166 | * number of targets.  |
167 | *  |
168 | * @section bibliography General Bibliography  |
169 | *  |
170 | * @see Abramowitz and Stegun: Handbook of Mathematical Functions,  |
171 | * with Formulas, Graphs, and Mathematical Tables  |
172 | * Edited by Milton Abramowitz and Irene A. Stegun,  |
173 | * National Bureau of Standards Applied Mathematics Series - 55  |
174 | * Issued June 1964, Tenth Printing, December 1972, with corrections  |
175 | * Electronic versions of A&S abound including both pdf and navigable html.  |
176 | * @see for example http://people.math.sfu.ca/~cbm/aands/  |
177 | *  |
178 | * @see The old A&S has been redone as the  |
179 | * NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/  |
180 | * This version is far more navigable and includes more recent work.  |
181 | *  |
182 | * @see An Atlas of Functions: with Equator, the Atlas Function Calculator  |
183 | * 2nd Edition, by Oldham, Keith B., Myland, Jan, Spanier, Jerome  |
184 | *  |
185 | * @see Asymptotics and Special Functions by Frank W. J. Olver,  |
186 | * Academic Press, 1974  |
187 | *  |
188 | * @see Numerical Recipes in C, The Art of Scientific Computing,  |
189 | * by William H. Press, Second Ed., Saul A. Teukolsky,  |
190 | * William T. Vetterling, and Brian P. Flannery,  |
191 | * Cambridge University Press, 1992  |
192 | *  |
193 | * @see The Special Functions and Their Approximations: Volumes 1 and 2,  |
194 | * by Yudell L. Luke, Academic Press, 1969  |
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$, for @c float parameters @c a, @c b.  |
308 | *  |
309 | * @see beta for more details.  |
310 | */  |
311 | inline float  |
312 | betaf(float __a, float __b)  |
313 | { return __detail::__beta<float>(__a, __b); }  |
314 |   |
315 | /**  |
316 | * Return the beta function, @f$B(a,b)@f$, for long double  |
317 | * parameters @c a, @c b.  |
318 | *  |
319 | * @see beta for more details.  |
320 | */  |
321 | inline long double  |
322 | betal(long double __a, long double __b)  |
323 | { return __detail::__beta<long double>(__a, __b); }  |
324 |   |
325 | /**  |
326 | * Return the beta function, @f$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$.  |
1180 | *  |
1181 | * The spherical Neumann function is defined by  |
1182 | * @f[  |
1183 | * n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x)  |
1184 | * @f]  |
1185 | *  |
1186 | * @tparam _Tp The floating-point type of the argument @c __x.  |
1187 | * @param __n The integral order <tt> n >= 0 </tt>  |
1188 | * @param __x The real argument <tt> __x >= 0 </tt>  |
1189 | * @throw std::domain_error if <tt> __x < 0 </tt>.  |
1190 | */  |
1191 | template<typename _Tp>  |
1192 | inline typename __gnu_cxx::__promote<_Tp>::__type  |
1193 | sph_neumann(unsigned int __n, _Tp __x)  |
1194 | {  |
1195 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type;  |
1196 | return __detail::__sph_neumann<__type>(__n, __x);  |
1197 | }  |
1198 |   |
1199 | // @} group mathsf  |
1200 |   |
1201 | _GLIBCXX_END_NAMESPACE_VERSION  |
1202 | } // namespace std  |
1203 |   |
1204 | #ifndef __STRICT_ANSI__  |
1205 | namespace __gnu_cxx _GLIBCXX_VISIBILITY(default)  |
1206 | {  |
1207 | _GLIBCXX_BEGIN_NAMESPACE_VERSION  |
1208 |   |
1209 | // Airy functions  |
1210 |   |
1211 | /**  |
1212 | * Return the Airy function @f$ Ai(x) @f$ of @c float argument x.  |
1213 | */  |
1214 | inline float  |
1215 | airy_aif(float __x)  |
1216 | {  |
1217 | float __Ai, __Bi, __Aip, __Bip;  |
1218 | std::__detail::__airy<float>(__x, __Ai, __Bi, __Aip, __Bip);  |
1219 | return __Ai;  |
1220 | }  |
1221 |   |
1222 | /**  |
1223 | * Return the Airy function @f$ Ai(x) @f$ of <tt>long double</tt> argument x.  |
1224 | */  |
1225 | inline long double  |
1226 | airy_ail(long double __x)  |
1227 | {  |
1228 | long double __Ai, __Bi, __Aip, __Bip;  |
1229 | std::__detail::__airy<long double>(__x, __Ai, __Bi, __Aip, __Bip);  |
1230 | return __Ai;  |
1231 | }  |
1232 |   |
1233 | /**  |
1234 | * Return the Airy function @f$ Ai(x) @f$ of real argument x.  |
1235 | */  |
1236 | template<typename _Tp>  |
1237 | inline typename __gnu_cxx::__promote<_Tp>::__type  |
1238 | airy_ai(_Tp __x)  |
1239 | {  |
1240 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type;  |
1241 | __type __Ai, __Bi, __Aip, __Bip;  |
1242 | std::__detail::__airy<__type>(__x, __Ai, __Bi, __Aip, __Bip);  |
1243 | return __Ai;  |
1244 | }  |
1245 |   |
1246 | /**  |
1247 | * Return the Airy function @f$ Bi(x) @f$ of @c float argument x.  |
1248 | */  |
1249 | inline float  |
1250 | airy_bif(float __x)  |
1251 | {  |
1252 | float __Ai, __Bi, __Aip, __Bip;  |
1253 | std::__detail::__airy<float>(__x, __Ai, __Bi, __Aip, __Bip);  |
1254 | return __Bi;  |
1255 | }  |
1256 |   |
1257 | /**  |
1258 | * Return the Airy function @f$ Bi(x) @f$ of <tt>long double</tt> argument x.  |
1259 | */  |
1260 | inline long double  |
1261 | airy_bil(long double __x)  |
1262 | {  |
1263 | long double __Ai, __Bi, __Aip, __Bip;  |
1264 | std::__detail::__airy<long double>(__x, __Ai, __Bi, __Aip, __Bip);  |
1265 | return __Bi;  |
1266 | }  |
1267 |   |
1268 | /**  |
1269 | * Return the Airy function @f$ Bi(x) @f$ of real argument x.  |
1270 | */  |
1271 | template<typename _Tp>  |
1272 | inline typename __gnu_cxx::__promote<_Tp>::__type  |
1273 | airy_bi(_Tp __x)  |
1274 | {  |
1275 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type;  |
1276 | __type __Ai, __Bi, __Aip, __Bip;  |
1277 | std::__detail::__airy<__type>(__x, __Ai, __Bi, __Aip, __Bip);  |
1278 | return __Bi;  |
1279 | }  |
1280 |   |
1281 | // Confluent hypergeometric functions  |
1282 |   |
1283 | /**  |
1284 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$  |
1285 | * of @c float numeratorial parameter @c a, denominatorial parameter @c c,  |
1286 | * and argument @c x.  |
1287 | *  |
1288 | * @see conf_hyperg for details.  |
1289 | */  |
1290 | inline float  |
1291 | conf_hypergf(float __a, float __c, float __x)  |
1292 | { return std::__detail::__conf_hyperg<float>(__a, __c, __x); }  |
1293 |   |
1294 | /**  |
1295 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$  |
1296 | * of <tt>long double</tt> numeratorial parameter @c a,  |
1297 | * denominatorial parameter @c c, and argument @c x.  |
1298 | *  |
1299 | * @see conf_hyperg for details.  |
1300 | */  |
1301 | inline long double  |
1302 | conf_hypergl(long double __a, long double __c, long double __x)  |
1303 | { return std::__detail::__conf_hyperg<long double>(__a, __c, __x); }  |
1304 |   |
1305 | /**  |
1306 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$  |
1307 | * of real numeratorial parameter @c a, denominatorial parameter @c c,  |
1308 | * and argument @c x.  |
1309 | *  |
1310 | * The confluent hypergeometric function is defined by  |
1311 | * @f[  |
1312 | * {}_1F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n x^n}{(c)_n n!}  |
1313 | * @f]  |
1314 | * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$,  |
1315 | * @f$ (x)_0 = 1 @f$  |
1316 | *  |
1317 | * @param __a The numeratorial parameter  |
1318 | * @param __c The denominatorial parameter  |
1319 | * @param __x The argument  |
1320 | */  |
1321 | template<typename _Tpa, typename _Tpc, typename _Tp>  |
1322 | inline typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type  |
1323 | conf_hyperg(_Tpa __a, _Tpc __c, _Tp __x)  |
1324 | {  |
1325 | typedef typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type __type;  |
1326 | return std::__detail::__conf_hyperg<__type>(__a, __c, __x);  |
1327 | }  |
1328 |   |
1329 | // Hypergeometric functions  |
1330 |   |
1331 | /**  |
1332 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$  |
1333 | * of @ float numeratorial parameters @c a and @c b,  |
1334 | * denominatorial parameter @c c, and argument @c x.  |
1335 | *  |
1336 | * @see hyperg for details.  |
1337 | */  |
1338 | inline float  |
1339 | hypergf(float __a, float __b, float __c, float __x)  |
1340 | { return std::__detail::__hyperg<float>(__a, __b, __c, __x); }  |
1341 |   |
1342 | /**  |
1343 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$  |
1344 | * of <tt>long double</tt> numeratorial parameters @c a and @c b,  |
1345 | * denominatorial parameter @c c, and argument @c x.  |
1346 | *  |
1347 | * @see hyperg for details.  |
1348 | */  |
1349 | inline long double  |
1350 | hypergl(long double __a, long double __b, long double __c, long double __x)  |
1351 | { return std::__detail::__hyperg<long double>(__a, __b, __c, __x); }  |
1352 |   |
1353 | /**  |
1354 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$  |
1355 | * of real numeratorial parameters @c a and @c b,  |
1356 | * denominatorial parameter @c c, and argument @c x.  |
1357 | *  |
1358 | * The hypergeometric function is defined by  |
1359 | * @f[  |
1360 | * {}_2F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n x^n}{(c)_n n!}  |
1361 | * @f]  |
1362 | * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$,  |
1363 | * @f$ (x)_0 = 1 @f$  |
1364 | *  |
1365 | * @param __a The first numeratorial parameter  |
1366 | * @param __b The second numeratorial parameter  |
1367 | * @param __c The denominatorial parameter  |
1368 | * @param __x The argument  |
1369 | */  |
1370 | template<typename _Tpa, typename _Tpb, typename _Tpc, typename _Tp>  |
1371 | inline typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp>::__type  |
1372 | hyperg(_Tpa __a, _Tpb __b, _Tpc __c, _Tp __x)  |
1373 | {  |
1374 | typedef typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp>  |
1375 | ::__type __type;  |
1376 | return std::__detail::__hyperg<__type>(__a, __b, __c, __x);  |
1377 | }  |
1378 |   |
1379 | _GLIBCXX_END_NAMESPACE_VERSION  |
1380 | } // namespace __gnu_cxx  |
1381 | #endif // __STRICT_ANSI__  |
1382 |   |
1383 | #pragma GCC visibility pop  |
1384 |   |
1385 | #endif // _GLIBCXX_BITS_SPECFUN_H  |
1386 | |