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1 | // Mathematical Special Functions for -*- C++ -*- |
2 | ||
cbe34bb5 | 3 | // Copyright (C) 2006-2017 Free Software Foundation, Inc. |
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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 | ||
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37 | #define __STDCPP_MATH_SPEC_FUNCS__ 201003L |
38 | ||
39 | #define __cpp_lib_math_special_functions 201603L | |
40 | ||
f8571e51 | 41 | #if __cplusplus <= 201403L && __STDCPP_WANT_MATH_SPEC_FUNCS__ == 0 |
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42 | # error include <cmath> and define __STDCPP_WANT_MATH_SPEC_FUNCS__ |
43 | #endif | |
44 | ||
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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 | ||
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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: | |
26bddba3 JW |
121 | * - @ref __gnu_cxx::conf_hyperg "conf_hyperg - Confluent hypergeometric functions" |
122 | * - @ref __gnu_cxx::hyperg "hyperg - Hypergeometric functions" | |
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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: | |
9a9534e1 | 129 | * 1. If any argument intended to be floating point is given an integral value |
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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 | ||
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197 | // Associated Laguerre polynomials |
198 | ||
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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 | */ | |
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205 | inline float |
206 | assoc_laguerref(unsigned int __n, unsigned int __m, float __x) | |
207 | { return __detail::__assoc_laguerre<float>(__n, __m, __x); } | |
208 | ||
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209 | /** |
210 | * Return the associated Laguerre polynomial of order @c n, | |
211 | * degree @c m: @f$ L_n^m(x) @f$. | |
212 | * | |
213 | * @see assoc_laguerre for more details. | |
214 | */ | |
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215 | inline long double |
216 | assoc_laguerrel(unsigned int __n, unsigned int __m, long double __x) | |
217 | { return __detail::__assoc_laguerre<long double>(__n, __m, __x); } | |
218 | ||
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219 | /** |
220 | * Return the associated Laguerre polynomial of nonnegative order @c n, | |
221 | * nonnegative degree @c m and real argument @c x: @f$ 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 | */ | |
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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 | ||
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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 | */ | |
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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 | ||
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270 | /** |
271 | * Return the associated Legendre function of degree @c l and order @c m. | |
272 | * | |
273 | * @see assoc_legendre for more details. | |
274 | */ | |
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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 | ||
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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 | */ | |
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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 | ||
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306 | /** |
307 | * Return the beta function, @f$ B(a,b) @f$, for @c float parameters @c a, @c b. | |
308 | * | |
309 | * @see beta for more details. | |
310 | */ | |
2be75957 | 311 | inline float |
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312 | betaf(float __a, float __b) |
313 | { return __detail::__beta<float>(__a, __b); } | |
2be75957 | 314 | |
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315 | /** |
316 | * Return the beta function, @f$B(a,b)@f$, for long double | |
317 | * parameters @c a, @c b. | |
318 | * | |
319 | * @see beta for more details. | |
320 | */ | |
2be75957 | 321 | inline long double |
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322 | betal(long double __a, long double __b) |
323 | { return __detail::__beta<long double>(__a, __b); } | |
2be75957 | 324 | |
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325 | /** |
326 | * Return the beta function, @f$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) | |
2be75957 | 344 | { |
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345 | typedef typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type __type; |
346 | return __detail::__beta<__type>(__a, __b); | |
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347 | } |
348 | ||
349 | // Complete elliptic integrals of the first kind | |
350 | ||
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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 | */ | |
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357 | inline float |
358 | comp_ellint_1f(float __k) | |
359 | { return __detail::__comp_ellint_1<float>(__k); } | |
360 | ||
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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 | */ | |
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367 | inline long double |
368 | comp_ellint_1l(long double __k) | |
369 | { return __detail::__comp_ellint_1<long double>(__k); } | |
370 | ||
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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$. | |
382 | * @see ellint_1 for details of the incomplete elliptic function | |
383 | * of the first kind. | |
384 | * | |
385 | * @tparam _Tp The floating-point type of the modulus @c __k. | |
386 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> | |
387 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. | |
388 | */ | |
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389 | template<typename _Tp> |
390 | inline typename __gnu_cxx::__promote<_Tp>::__type | |
391 | comp_ellint_1(_Tp __k) | |
392 | { | |
393 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; | |
394 | return __detail::__comp_ellint_1<__type>(__k); | |
395 | } | |
396 | ||
397 | // Complete elliptic integrals of the second kind | |
398 | ||
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399 | /** |
400 | * Return the complete elliptic integral of the second kind @f$ E(k) @f$ | |
401 | * for @c float modulus @c k. | |
402 | * | |
403 | * @see comp_ellint_2 for details. | |
404 | */ | |
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405 | inline float |
406 | comp_ellint_2f(float __k) | |
407 | { return __detail::__comp_ellint_2<float>(__k); } | |
408 | ||
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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 | */ | |
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415 | inline long double |
416 | comp_ellint_2l(long double __k) | |
417 | { return __detail::__comp_ellint_2<long double>(__k); } | |
418 | ||
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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$. | |
429 | * @see ellint_2 for details of the incomplete elliptic function | |
430 | * of the second kind. | |
431 | * | |
432 | * @tparam _Tp The floating-point type of the modulus @c __k. | |
433 | * @param __k The modulus, @c abs(__k) <= 1 | |
434 | * @throw std::domain_error if @c abs(__k) > 1. | |
435 | */ | |
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436 | template<typename _Tp> |
437 | inline typename __gnu_cxx::__promote<_Tp>::__type | |
438 | comp_ellint_2(_Tp __k) | |
439 | { | |
440 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; | |
441 | return __detail::__comp_ellint_2<__type>(__k); | |
442 | } | |
443 | ||
444 | // Complete elliptic integrals of the third kind | |
445 | ||
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446 | /** |
447 | * @brief Return the complete elliptic integral of the third kind | |
448 | * @f$ \Pi(k,\nu) @f$ for @c float modulus @c k. | |
449 | * | |
450 | * @see comp_ellint_3 for details. | |
451 | */ | |
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452 | inline float |
453 | comp_ellint_3f(float __k, float __nu) | |
454 | { return __detail::__comp_ellint_3<float>(__k, __nu); } | |
455 | ||
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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 | */ | |
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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 | ||
0c39f36d ESR |
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$. | |
478 | * @see ellint_3 for details of the incomplete elliptic function | |
479 | * of the third kind. | |
480 | * | |
481 | * @tparam _Tp The floating-point type of the modulus @c __k. | |
482 | * @tparam _Tpn The floating-point type of the argument @c __nu. | |
483 | * @param __k The modulus, @c abs(__k) <= 1 | |
484 | * @param __nu The argument | |
485 | * @throw std::domain_error if @c abs(__k) > 1. | |
486 | */ | |
2be75957 ESR |
487 | template<typename _Tp, typename _Tpn> |
488 | inline typename __gnu_cxx::__promote_2<_Tp, _Tpn>::__type | |
489 | comp_ellint_3(_Tp __k, _Tpn __nu) | |
490 | { | |
491 | typedef typename __gnu_cxx::__promote_2<_Tp, _Tpn>::__type __type; | |
492 | return __detail::__comp_ellint_3<__type>(__k, __nu); | |
493 | } | |
494 | ||
495 | // Regular modified cylindrical Bessel functions | |
496 | ||
0c39f36d ESR |
497 | /** |
498 | * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ | |
499 | * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. | |
500 | * | |
501 | * @see cyl_bessel_i for setails. | |
502 | */ | |
2be75957 ESR |
503 | inline float |
504 | cyl_bessel_if(float __nu, float __x) | |
505 | { return __detail::__cyl_bessel_i<float>(__nu, __x); } | |
506 | ||
0c39f36d ESR |
507 | /** |
508 | * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ | |
509 | * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. | |
510 | * | |
511 | * @see cyl_bessel_i for setails. | |
512 | */ | |
2be75957 ESR |
513 | inline long double |
514 | cyl_bessel_il(long double __nu, long double __x) | |
515 | { return __detail::__cyl_bessel_i<long double>(__nu, __x); } | |
516 | ||
0c39f36d ESR |
517 | /** |
518 | * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ | |
519 | * for real order @f$ \nu @f$ and argument @f$ x >= 0 @f$. | |
520 | * | |
521 | * The regular modified cylindrical Bessel function is: | |
522 | * @f[ | |
523 | * I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty} | |
524 | * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} | |
525 | * @f] | |
526 | * | |
527 | * @tparam _Tpnu The floating-point type of the order @c __nu. | |
528 | * @tparam _Tp The floating-point type of the argument @c __x. | |
529 | * @param __nu The order | |
530 | * @param __x The argument, <tt> __x >= 0 </tt> | |
531 | * @throw std::domain_error if <tt> __x < 0 </tt>. | |
532 | */ | |
2be75957 ESR |
533 | template<typename _Tpnu, typename _Tp> |
534 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type | |
535 | cyl_bessel_i(_Tpnu __nu, _Tp __x) | |
536 | { | |
537 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; | |
538 | return __detail::__cyl_bessel_i<__type>(__nu, __x); | |
539 | } | |
540 | ||
541 | // Cylindrical Bessel functions (of the first kind) | |
542 | ||
0c39f36d ESR |
543 | /** |
544 | * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$ | |
545 | * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. | |
546 | * | |
547 | * @see cyl_bessel_j for setails. | |
548 | */ | |
2be75957 ESR |
549 | inline float |
550 | cyl_bessel_jf(float __nu, float __x) | |
551 | { return __detail::__cyl_bessel_j<float>(__nu, __x); } | |
552 | ||
0c39f36d ESR |
553 | /** |
554 | * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$ | |
555 | * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. | |
556 | * | |
557 | * @see cyl_bessel_j for setails. | |
558 | */ | |
2be75957 ESR |
559 | inline long double |
560 | cyl_bessel_jl(long double __nu, long double __x) | |
561 | { return __detail::__cyl_bessel_j<long double>(__nu, __x); } | |
562 | ||
0c39f36d ESR |
563 | /** |
564 | * Return the Bessel function @f$ J_{\nu}(x) @f$ of real order @f$ \nu @f$ | |
565 | * and argument @f$ x >= 0 @f$. | |
566 | * | |
567 | * The cylindrical Bessel function is: | |
568 | * @f[ | |
569 | * J_{\nu}(x) = \sum_{k=0}^{\infty} | |
570 | * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} | |
571 | * @f] | |
572 | * | |
573 | * @tparam _Tpnu The floating-point type of the order @c __nu. | |
574 | * @tparam _Tp The floating-point type of the argument @c __x. | |
575 | * @param __nu The order | |
576 | * @param __x The argument, <tt> __x >= 0 </tt> | |
577 | * @throw std::domain_error if <tt> __x < 0 </tt>. | |
578 | */ | |
2be75957 ESR |
579 | template<typename _Tpnu, typename _Tp> |
580 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type | |
581 | cyl_bessel_j(_Tpnu __nu, _Tp __x) | |
582 | { | |
583 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; | |
584 | return __detail::__cyl_bessel_j<__type>(__nu, __x); | |
585 | } | |
586 | ||
587 | // Irregular modified cylindrical Bessel functions | |
588 | ||
0c39f36d ESR |
589 | /** |
590 | * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ | |
591 | * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. | |
592 | * | |
593 | * @see cyl_bessel_k for setails. | |
594 | */ | |
2be75957 ESR |
595 | inline float |
596 | cyl_bessel_kf(float __nu, float __x) | |
597 | { return __detail::__cyl_bessel_k<float>(__nu, __x); } | |
598 | ||
0c39f36d ESR |
599 | /** |
600 | * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ | |
601 | * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. | |
602 | * | |
603 | * @see cyl_bessel_k for setails. | |
604 | */ | |
2be75957 ESR |
605 | inline long double |
606 | cyl_bessel_kl(long double __nu, long double __x) | |
607 | { return __detail::__cyl_bessel_k<long double>(__nu, __x); } | |
608 | ||
0c39f36d ESR |
609 | /** |
610 | * Return the irregular modified Bessel function @f$ 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$. | |
620 | * For negative argument we have simply: | |
621 | * @f[ | |
622 | * K_{-\nu}(x) = K_{\nu}(x) | |
623 | * @f] | |
624 | * | |
625 | * @tparam _Tpnu The floating-point type of the order @c __nu. | |
626 | * @tparam _Tp The floating-point type of the argument @c __x. | |
627 | * @param __nu The order | |
628 | * @param __x The argument, <tt> __x >= 0 </tt> | |
629 | * @throw std::domain_error if <tt> __x < 0 </tt>. | |
630 | */ | |
2be75957 ESR |
631 | template<typename _Tpnu, typename _Tp> |
632 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type | |
633 | cyl_bessel_k(_Tpnu __nu, _Tp __x) | |
634 | { | |
635 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; | |
636 | return __detail::__cyl_bessel_k<__type>(__nu, __x); | |
637 | } | |
638 | ||
639 | // Cylindrical Neumann functions | |
640 | ||
0c39f36d ESR |
641 | /** |
642 | * Return the Neumann function @f$ N_{\nu}(x) @f$ | |
643 | * of @c float order @f$ \nu @f$ and argument @f$ x @f$. | |
644 | * | |
645 | * @see cyl_neumann for setails. | |
646 | */ | |
2be75957 ESR |
647 | inline float |
648 | cyl_neumannf(float __nu, float __x) | |
649 | { return __detail::__cyl_neumann_n<float>(__nu, __x); } | |
650 | ||
0c39f36d ESR |
651 | /** |
652 | * Return the Neumann function @f$ N_{\nu}(x) @f$ | |
653 | * of <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x @f$. | |
654 | * | |
655 | * @see cyl_neumann for setails. | |
656 | */ | |
2be75957 ESR |
657 | inline long double |
658 | cyl_neumannl(long double __nu, long double __x) | |
659 | { return __detail::__cyl_neumann_n<long double>(__nu, __x); } | |
660 | ||
0c39f36d ESR |
661 | /** |
662 | * Return the Neumann function @f$ 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$. | |
672 | * | |
673 | * @tparam _Tpnu The floating-point type of the order @c __nu. | |
674 | * @tparam _Tp The floating-point type of the argument @c __x. | |
675 | * @param __nu The order | |
676 | * @param __x The argument, <tt> __x >= 0 </tt> | |
677 | * @throw std::domain_error if <tt> __x < 0 </tt>. | |
678 | */ | |
2be75957 ESR |
679 | template<typename _Tpnu, typename _Tp> |
680 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type | |
681 | cyl_neumann(_Tpnu __nu, _Tp __x) | |
682 | { | |
683 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; | |
684 | return __detail::__cyl_neumann_n<__type>(__nu, __x); | |
685 | } | |
686 | ||
687 | // Incomplete elliptic integrals of the first kind | |
688 | ||
0c39f36d ESR |
689 | /** |
690 | * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$ | |
691 | * for @c float modulus @f$ k @f$ and angle @f$ \phi @f$. | |
692 | * | |
693 | * @see ellint_1 for details. | |
694 | */ | |
2be75957 ESR |
695 | inline float |
696 | ellint_1f(float __k, float __phi) | |
697 | { return __detail::__ellint_1<float>(__k, __phi); } | |
698 | ||
0c39f36d ESR |
699 | /** |
700 | * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$ | |
701 | * for <tt>long double</tt> modulus @f$ k @f$ and angle @f$ \phi @f$. | |
702 | * | |
703 | * @see ellint_1 for details. | |
704 | */ | |
2be75957 ESR |
705 | inline long double |
706 | ellint_1l(long double __k, long double __phi) | |
707 | { return __detail::__ellint_1<long double>(__k, __phi); } | |
708 | ||
0c39f36d ESR |
709 | /** |
710 | * Return the incomplete elliptic integral of the first kind @f$ 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$. @see comp_ellint_1. | |
720 | * | |
721 | * @tparam _Tp The floating-point type of the modulus @c __k. | |
722 | * @tparam _Tpp The floating-point type of the angle @c __phi. | |
723 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> | |
724 | * @param __phi The integral limit argument in radians | |
725 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. | |
726 | */ | |
2be75957 ESR |
727 | template<typename _Tp, typename _Tpp> |
728 | inline typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type | |
729 | ellint_1(_Tp __k, _Tpp __phi) | |
730 | { | |
731 | typedef typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type __type; | |
732 | return __detail::__ellint_1<__type>(__k, __phi); | |
733 | } | |
734 | ||
735 | // Incomplete elliptic integrals of the second kind | |
736 | ||
0c39f36d ESR |
737 | /** |
738 | * @brief Return the incomplete elliptic integral of the second kind | |
739 | * @f$ E(k,\phi) @f$ for @c float argument. | |
740 | * | |
741 | * @see ellint_2 for details. | |
742 | */ | |
2be75957 ESR |
743 | inline float |
744 | ellint_2f(float __k, float __phi) | |
745 | { return __detail::__ellint_2<float>(__k, __phi); } | |
746 | ||
0c39f36d ESR |
747 | /** |
748 | * @brief Return the incomplete elliptic integral of the second kind | |
749 | * @f$ E(k,\phi) @f$. | |
750 | * | |
751 | * @see ellint_2 for details. | |
752 | */ | |
2be75957 ESR |
753 | inline long double |
754 | ellint_2l(long double __k, long double __phi) | |
755 | { return __detail::__ellint_2<long double>(__k, __phi); } | |
756 | ||
0c39f36d ESR |
757 | /** |
758 | * Return the incomplete elliptic integral of the second kind | |
759 | * @f$ 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$. @see comp_ellint_2. | |
767 | * | |
768 | * @tparam _Tp The floating-point type of the modulus @c __k. | |
769 | * @tparam _Tpp The floating-point type of the angle @c __phi. | |
770 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> | |
771 | * @param __phi The integral limit argument in radians | |
772 | * @return The elliptic function of the second kind. | |
773 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. | |
774 | */ | |
2be75957 ESR |
775 | template<typename _Tp, typename _Tpp> |
776 | inline typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type | |
777 | ellint_2(_Tp __k, _Tpp __phi) | |
778 | { | |
779 | typedef typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type __type; | |
780 | return __detail::__ellint_2<__type>(__k, __phi); | |
781 | } | |
782 | ||
783 | // Incomplete elliptic integrals of the third kind | |
784 | ||
0c39f36d ESR |
785 | /** |
786 | * @brief Return the incomplete elliptic integral of the third kind | |
787 | * @f$ \Pi(k,\nu,\phi) @f$ for @c float argument. | |
788 | * | |
789 | * @see ellint_3 for details. | |
790 | */ | |
2be75957 ESR |
791 | inline float |
792 | ellint_3f(float __k, float __nu, float __phi) | |
793 | { return __detail::__ellint_3<float>(__k, __nu, __phi); } | |
794 | ||
0c39f36d ESR |
795 | /** |
796 | * @brief Return the incomplete elliptic integral of the third kind | |
797 | * @f$ \Pi(k,\nu,\phi) @f$. | |
798 | * | |
799 | * @see ellint_3 for details. | |
800 | */ | |
2be75957 ESR |
801 | inline long double |
802 | ellint_3l(long double __k, long double __nu, long double __phi) | |
803 | { return __detail::__ellint_3<long double>(__k, __nu, __phi); } | |
804 | ||
0c39f36d ESR |
805 | /** |
806 | * @brief Return the incomplete elliptic integral of the third kind | |
807 | * @f$ \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$. @see comp_ellint_3. | |
818 | * | |
819 | * @tparam _Tp The floating-point type of the modulus @c __k. | |
820 | * @tparam _Tpn The floating-point type of the argument @c __nu. | |
821 | * @tparam _Tpp The floating-point type of the angle @c __phi. | |
822 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> | |
823 | * @param __nu The second argument | |
824 | * @param __phi The integral limit argument in radians | |
825 | * @return The elliptic function of the third kind. | |
826 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. | |
827 | */ | |
2be75957 ESR |
828 | template<typename _Tp, typename _Tpn, typename _Tpp> |
829 | inline typename __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type | |
830 | ellint_3(_Tp __k, _Tpn __nu, _Tpp __phi) | |
831 | { | |
832 | typedef typename __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type __type; | |
833 | return __detail::__ellint_3<__type>(__k, __nu, __phi); | |
834 | } | |
835 | ||
836 | // Exponential integrals | |
837 | ||
0c39f36d ESR |
838 | /** |
839 | * Return the exponential integral @f$ Ei(x) @f$ for @c float argument @c x. | |
840 | * | |
841 | * @see expint for details. | |
842 | */ | |
2be75957 ESR |
843 | inline float |
844 | expintf(float __x) | |
845 | { return __detail::__expint<float>(__x); } | |
846 | ||
0c39f36d ESR |
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 | */ | |
2be75957 ESR |
853 | inline long double |
854 | expintl(long double __x) | |
855 | { return __detail::__expint<long double>(__x); } | |
856 | ||
0c39f36d ESR |
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 | */ | |
2be75957 ESR |
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 | ||
0c39f36d ESR |
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 | */ | |
2be75957 ESR |
884 | inline float |
885 | hermitef(unsigned int __n, float __x) | |
886 | { return __detail::__poly_hermite<float>(__n, __x); } | |
887 | ||
0c39f36d ESR |
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 | */ | |
2be75957 ESR |
894 | inline long double |
895 | hermitel(unsigned int __n, long double __x) | |
896 | { return __detail::__poly_hermite<long double>(__n, __x); } | |
897 | ||
0c39f36d ESR |
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 | */ | |
2be75957 ESR |
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 | ||
0c39f36d ESR |
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$. | |
929 | * | |
930 | * @see laguerre for more details. | |
931 | */ | |
2be75957 ESR |
932 | inline float |
933 | laguerref(unsigned int __n, float __x) | |
934 | { return __detail::__laguerre<float>(__n, __x); } | |
935 | ||
0c39f36d ESR |
936 | /** |
937 | * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n | |
938 | * and <tt>long double</tt> argument @f$ x >= 0 @f$. | |
939 | * | |
940 | * @see laguerre for more details. | |
941 | */ | |
2be75957 ESR |
942 | inline long double |
943 | laguerrel(unsigned int __n, long double __x) | |
944 | { return __detail::__laguerre<long double>(__n, __x); } | |
945 | ||
0c39f36d ESR |
946 | /** |
947 | * Returns the Laguerre polynomial @f$ L_n(x) @f$ | |
948 | * of nonnegative degree @c n and real argument @f$ x >= 0 @f$. | |
949 | * | |
950 | * The Laguerre polynomial is defined by: | |
951 | * @f[ | |
952 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) | |
953 | * @f] | |
954 | * | |
955 | * @tparam _Tp The floating-point type of the argument @c __x. | |
956 | * @param __n The nonnegative order | |
957 | * @param __x The argument <tt> __x >= 0 </tt> | |
958 | * @throw std::domain_error if <tt> __x < 0 </tt>. | |
959 | */ | |
2be75957 ESR |
960 | template<typename _Tp> |
961 | inline typename __gnu_cxx::__promote<_Tp>::__type | |
962 | laguerre(unsigned int __n, _Tp __x) | |
963 | { | |
964 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; | |
965 | return __detail::__laguerre<__type>(__n, __x); | |
966 | } | |
967 | ||
968 | // Legendre polynomials | |
969 | ||
0c39f36d ESR |
970 | /** |
971 | * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative | |
972 | * degree @f$ l @f$ and @c float argument @f$ |x| <= 0 @f$. | |
973 | * | |
974 | * @see legendre for more details. | |
975 | */ | |
2be75957 | 976 | inline float |
0c39f36d ESR |
977 | legendref(unsigned int __l, float __x) |
978 | { return __detail::__poly_legendre_p<float>(__l, __x); } | |
2be75957 | 979 | |
0c39f36d ESR |
980 | /** |
981 | * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative | |
982 | * degree @f$ l @f$ and <tt>long double</tt> argument @f$ |x| <= 0 @f$. | |
983 | * | |
984 | * @see legendre for more details. | |
985 | */ | |
2be75957 | 986 | inline long double |
0c39f36d ESR |
987 | legendrel(unsigned int __l, long double __x) |
988 | { return __detail::__poly_legendre_p<long double>(__l, __x); } | |
2be75957 | 989 | |
0c39f36d ESR |
990 | /** |
991 | * Return the Legendre polynomial @f$ 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 | */ | |
2be75957 ESR |
1005 | template<typename _Tp> |
1006 | inline typename __gnu_cxx::__promote<_Tp>::__type | |
0c39f36d | 1007 | legendre(unsigned int __l, _Tp __x) |
2be75957 ESR |
1008 | { |
1009 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; | |
0c39f36d | 1010 | return __detail::__poly_legendre_p<__type>(__l, __x); |
2be75957 ESR |
1011 | } |
1012 | ||
1013 | // Riemann zeta functions | |
1014 | ||
0c39f36d ESR |
1015 | /** |
1016 | * Return the Riemann zeta function @f$ \zeta(s) @f$ | |
1017 | * for @c float argument @f$ s @f$. | |
1018 | * | |
1019 | * @see riemann_zeta for more details. | |
1020 | */ | |
2be75957 ESR |
1021 | inline float |
1022 | riemann_zetaf(float __s) | |
1023 | { return __detail::__riemann_zeta<float>(__s); } | |
1024 | ||
0c39f36d ESR |
1025 | /** |
1026 | * Return the Riemann zeta function @f$ \zeta(s) @f$ | |
1027 | * for <tt>long double</tt> argument @f$ s @f$. | |
1028 | * | |
1029 | * @see riemann_zeta for more details. | |
1030 | */ | |
2be75957 ESR |
1031 | inline long double |
1032 | riemann_zetal(long double __s) | |
1033 | { return __detail::__riemann_zeta<long double>(__s); } | |
1034 | ||
0c39f36d ESR |
1035 | /** |
1036 | * Return the Riemann zeta function @f$ \zeta(s) @f$ | |
1037 | * for real argument @f$ s @f$. | |
1038 | * | |
1039 | * The Riemann zeta function is defined by: | |
1040 | * @f[ | |
1041 | * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1 | |
1042 | * @f] | |
1043 | * and | |
1044 | * @f[ | |
1045 | * \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s} | |
1046 | * \hbox{ for } 0 <= s <= 1 | |
1047 | * @f] | |
1048 | * For s < 1 use the reflection formula: | |
1049 | * @f[ | |
1050 | * \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s) | |
1051 | * @f] | |
1052 | * | |
1053 | * @tparam _Tp The floating-point type of the argument @c __s. | |
1054 | * @param __s The argument <tt> s != 1 </tt> | |
1055 | */ | |
2be75957 ESR |
1056 | template<typename _Tp> |
1057 | inline typename __gnu_cxx::__promote<_Tp>::__type | |
1058 | riemann_zeta(_Tp __s) | |
1059 | { | |
1060 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; | |
1061 | return __detail::__riemann_zeta<__type>(__s); | |
1062 | } | |
1063 | ||
1064 | // Spherical Bessel functions | |
1065 | ||
0c39f36d ESR |
1066 | /** |
1067 | * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n | |
1068 | * and @c float argument @f$ x >= 0 @f$. | |
1069 | * | |
1070 | * @see sph_bessel for more details. | |
1071 | */ | |
2be75957 ESR |
1072 | inline float |
1073 | sph_besself(unsigned int __n, float __x) | |
1074 | { return __detail::__sph_bessel<float>(__n, __x); } | |
1075 | ||
0c39f36d ESR |
1076 | /** |
1077 | * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n | |
1078 | * and <tt>long double</tt> argument @f$ x >= 0 @f$. | |
1079 | * | |
1080 | * @see sph_bessel for more details. | |
1081 | */ | |
2be75957 ESR |
1082 | inline long double |
1083 | sph_bessell(unsigned int __n, long double __x) | |
1084 | { return __detail::__sph_bessel<long double>(__n, __x); } | |
1085 | ||
0c39f36d ESR |
1086 | /** |
1087 | * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n | |
1088 | * and real argument @f$ x >= 0 @f$. | |
1089 | * | |
1090 | * The spherical Bessel function is defined by: | |
1091 | * @f[ | |
1092 | * j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) | |
1093 | * @f] | |
1094 | * | |
1095 | * @tparam _Tp The floating-point type of the argument @c __x. | |
1096 | * @param __n The integral order <tt> n >= 0 </tt> | |
1097 | * @param __x The real argument <tt> x >= 0 </tt> | |
1098 | * @throw std::domain_error if <tt> __x < 0 </tt>. | |
1099 | */ | |
2be75957 ESR |
1100 | template<typename _Tp> |
1101 | inline typename __gnu_cxx::__promote<_Tp>::__type | |
1102 | sph_bessel(unsigned int __n, _Tp __x) | |
1103 | { | |
1104 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; | |
1105 | return __detail::__sph_bessel<__type>(__n, __x); | |
1106 | } | |
1107 | ||
1108 | // Spherical associated Legendre functions | |
1109 | ||
0c39f36d ESR |
1110 | /** |
1111 | * Return the spherical Legendre function of nonnegative integral | |
1112 | * degree @c l and order @c m and float angle @f$ \theta @f$ in radians. | |
1113 | * | |
1114 | * @see sph_legendre for details. | |
1115 | */ | |
2be75957 ESR |
1116 | inline float |
1117 | sph_legendref(unsigned int __l, unsigned int __m, float __theta) | |
1118 | { return __detail::__sph_legendre<float>(__l, __m, __theta); } | |
1119 | ||
0c39f36d ESR |
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 | */ | |
2be75957 ESR |
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 | ||
0c39f36d ESR |
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 | */ | |
2be75957 ESR |
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 | ||
0c39f36d ESR |
1157 | /** |
1158 | * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ | |
1159 | * and @c float argument @f$ x >= 0 @f$. | |
1160 | * | |
1161 | * @see sph_neumann for details. | |
1162 | */ | |
2be75957 ESR |
1163 | inline float |
1164 | sph_neumannf(unsigned int __n, float __x) | |
1165 | { return __detail::__sph_neumann<float>(__n, __x); } | |
1166 | ||
0c39f36d ESR |
1167 | /** |
1168 | * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ | |
1169 | * and <tt>long double</tt> @f$ x >= 0 @f$. | |
1170 | * | |
1171 | * @see sph_neumann for details. | |
1172 | */ | |
2be75957 ESR |
1173 | inline long double |
1174 | sph_neumannl(unsigned int __n, long double __x) | |
1175 | { return __detail::__sph_neumann<long double>(__n, __x); } | |
1176 | ||
0c39f36d ESR |
1177 | /** |
1178 | * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ | |
1179 | * and real argument @f$ x >= 0 @f$. | |
1180 | * | |
1181 | * The spherical Neumann function is defined by | |
1182 | * @f[ | |
1183 | * n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) | |
1184 | * @f] | |
1185 | * | |
1186 | * @tparam _Tp The floating-point type of the argument @c __x. | |
1187 | * @param __n The integral order <tt> n >= 0 </tt> | |
1188 | * @param __x The real argument <tt> __x >= 0 </tt> | |
1189 | * @throw std::domain_error if <tt> __x < 0 </tt>. | |
1190 | */ | |
2be75957 ESR |
1191 | template<typename _Tp> |
1192 | inline typename __gnu_cxx::__promote<_Tp>::__type | |
1193 | sph_neumann(unsigned int __n, _Tp __x) | |
1194 | { | |
1195 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; | |
1196 | return __detail::__sph_neumann<__type>(__n, __x); | |
1197 | } | |
1198 | ||
1199 | // @} group mathsf | |
1200 | ||
1201 | _GLIBCXX_END_NAMESPACE_VERSION | |
1202 | } // namespace std | |
1203 | ||
1204 | namespace __gnu_cxx _GLIBCXX_VISIBILITY(default) | |
1205 | { | |
4a15d842 | 1206 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
2be75957 ESR |
1207 | |
1208 | // Confluent hypergeometric functions | |
1209 | ||
0c39f36d ESR |
1210 | /** |
1211 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ | |
1212 | * of @c float numeratorial parameter @c a, denominatorial parameter @c c, | |
1213 | * and argument @c x. | |
1214 | * | |
1215 | * @see conf_hyperg for details. | |
1216 | */ | |
2be75957 ESR |
1217 | inline float |
1218 | conf_hypergf(float __a, float __c, float __x) | |
1219 | { return std::__detail::__conf_hyperg<float>(__a, __c, __x); } | |
1220 | ||
0c39f36d ESR |
1221 | /** |
1222 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ | |
1223 | * of <tt>long double</tt> numeratorial parameter @c a, | |
1224 | * denominatorial parameter @c c, and argument @c x. | |
1225 | * | |
1226 | * @see conf_hyperg for details. | |
1227 | */ | |
2be75957 ESR |
1228 | inline long double |
1229 | conf_hypergl(long double __a, long double __c, long double __x) | |
1230 | { return std::__detail::__conf_hyperg<long double>(__a, __c, __x); } | |
1231 | ||
0c39f36d ESR |
1232 | /** |
1233 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ | |
1234 | * of real numeratorial parameter @c a, denominatorial parameter @c c, | |
1235 | * and argument @c x. | |
1236 | * | |
1237 | * The confluent hypergeometric function is defined by | |
1238 | * @f[ | |
1239 | * {}_1F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n x^n}{(c)_n n!} | |
1240 | * @f] | |
1241 | * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$, | |
1242 | * @f$ (x)_0 = 1 @f$ | |
1243 | * | |
1244 | * @param __a The numeratorial parameter | |
1245 | * @param __c The denominatorial parameter | |
1246 | * @param __x The argument | |
1247 | */ | |
2be75957 ESR |
1248 | template<typename _Tpa, typename _Tpc, typename _Tp> |
1249 | inline typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type | |
1250 | conf_hyperg(_Tpa __a, _Tpc __c, _Tp __x) | |
1251 | { | |
1252 | typedef typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type __type; | |
1253 | return std::__detail::__conf_hyperg<__type>(__a, __c, __x); | |
1254 | } | |
1255 | ||
1256 | // Hypergeometric functions | |
1257 | ||
0c39f36d ESR |
1258 | /** |
1259 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ | |
1260 | * of @ float numeratorial parameters @c a and @c b, | |
1261 | * denominatorial parameter @c c, and argument @c x. | |
1262 | * | |
1263 | * @see hyperg for details. | |
1264 | */ | |
2be75957 ESR |
1265 | inline float |
1266 | hypergf(float __a, float __b, float __c, float __x) | |
1267 | { return std::__detail::__hyperg<float>(__a, __b, __c, __x); } | |
1268 | ||
0c39f36d ESR |
1269 | /** |
1270 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ | |
1271 | * of <tt>long double</tt> numeratorial parameters @c a and @c b, | |
1272 | * denominatorial parameter @c c, and argument @c x. | |
1273 | * | |
1274 | * @see hyperg for details. | |
1275 | */ | |
2be75957 ESR |
1276 | inline long double |
1277 | hypergl(long double __a, long double __b, long double __c, long double __x) | |
1278 | { return std::__detail::__hyperg<long double>(__a, __b, __c, __x); } | |
1279 | ||
0c39f36d ESR |
1280 | /** |
1281 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ | |
1282 | * of real numeratorial parameters @c a and @c b, | |
1283 | * denominatorial parameter @c c, and argument @c x. | |
1284 | * | |
1285 | * The hypergeometric function is defined by | |
1286 | * @f[ | |
1287 | * {}_2F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n x^n}{(c)_n n!} | |
1288 | * @f] | |
1289 | * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$, | |
1290 | * @f$ (x)_0 = 1 @f$ | |
1291 | * | |
1292 | * @param __a The first numeratorial parameter | |
1293 | * @param __b The second numeratorial parameter | |
1294 | * @param __c The denominatorial parameter | |
1295 | * @param __x The argument | |
1296 | */ | |
2be75957 ESR |
1297 | template<typename _Tpa, typename _Tpb, typename _Tpc, typename _Tp> |
1298 | inline typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp>::__type | |
1299 | hyperg(_Tpa __a, _Tpb __b, _Tpc __c, _Tp __x) | |
1300 | { | |
1301 | typedef typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp> | |
1302 | ::__type __type; | |
1303 | return std::__detail::__hyperg<__type>(__a, __b, __c, __x); | |
1304 | } | |
1305 | ||
4a15d842 | 1306 | _GLIBCXX_END_NAMESPACE_VERSION |
2be75957 ESR |
1307 | } // namespace __gnu_cxx |
1308 | ||
1309 | #pragma GCC visibility pop | |
1310 | ||
1311 | #endif // _GLIBCXX_BITS_SPECFUN_H |