| 1 | // Mathematical Special Functions for -*- C++ -*- |
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| 2 | |
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| 3 | // Copyright (C) 2006-2021 Free Software Foundation, Inc. |
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| 4 | // |
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| 5 | // This file is part of the GNU ISO C++ Library. This library is free |
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| 6 | // software; you can redistribute it and/or modify it under the |
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| 7 | // terms of the GNU General Public License as published by the |
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| 8 | // Free Software Foundation; either version 3, or (at your option) |
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| 9 | // any later version. |
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| 10 | |
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| 11 | // This library is distributed in the hope that it will be useful, |
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| 12 | // but WITHOUT ANY WARRANTY; without even the implied warranty of |
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| 13 | // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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| 14 | // GNU General Public License for more details. |
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| 15 | |
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| 16 | // Under Section 7 of GPL version 3, you are granted additional |
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| 17 | // permissions described in the GCC Runtime Library Exception, version |
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| 18 | // 3.1, as published by the Free Software Foundation. |
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| 19 | |
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| 20 | // You should have received a copy of the GNU General Public License and |
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| 21 | // a copy of the GCC Runtime Library Exception along with this program; |
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| 22 | // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see |
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| 23 | // <http://www.gnu.org/licenses/>. |
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| 24 | |
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| 25 | /** @file bits/specfun.h |
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| 26 | * This is an internal header file, included by other library headers. |
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| 27 | * Do not attempt to use it directly. @headername{cmath} |
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| 28 | */ |
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| 29 | |
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| 30 | #ifndef _GLIBCXX_BITS_SPECFUN_H |
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| 31 | #define _GLIBCXX_BITS_SPECFUN_H 1 |
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| 32 | |
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| 33 | #pragma GCC visibility push(default) |
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| 34 | |
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| 35 | #include <bits/c++config.h> |
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| 36 | |
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| 37 | #define __STDCPP_MATH_SPEC_FUNCS__ 201003L |
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| 38 | |
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| 39 | #define __cpp_lib_math_special_functions 201603L |
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| 40 | |
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| 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__ |
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| 43 | #endif |
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| 44 | |
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| 45 | #include <bits/stl_algobase.h> |
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| 46 | #include <limits> |
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| 47 | #include <type_traits> |
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| 48 | |
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| 49 | #include <tr1/gamma.tcc> |
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| 50 | #include <tr1/bessel_function.tcc> |
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| 51 | #include <tr1/beta_function.tcc> |
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| 52 | #include <tr1/ell_integral.tcc> |
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| 53 | #include <tr1/exp_integral.tcc> |
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| 54 | #include <tr1/hypergeometric.tcc> |
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| 55 | #include <tr1/legendre_function.tcc> |
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| 56 | #include <tr1/modified_bessel_func.tcc> |
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| 57 | #include <tr1/poly_hermite.tcc> |
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| 58 | #include <tr1/poly_laguerre.tcc> |
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| 59 | #include <tr1/riemann_zeta.tcc> |
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| 60 | |
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| 61 | namespace std _GLIBCXX_VISIBILITY(default) |
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| 62 | { |
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| 63 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
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| 64 | |
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| 65 | /** |
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| 66 | * @defgroup mathsf Mathematical Special Functions |
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| 67 | * @ingroup numerics |
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| 68 | * |
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| 69 | * @section mathsf_desc Mathematical Special Functions |
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| 70 | * |
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| 71 | * A collection of advanced mathematical special functions, |
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| 72 | * defined by ISO/IEC IS 29124 and then added to ISO C++ 2017. |
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| 73 | * |
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| 74 | * |
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| 75 | * @subsection mathsf_intro Introduction and History |
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| 76 | * The first significant library upgrade on the road to C++2011, |
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| 77 | * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2005/n1836.pdf"> |
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| 78 | * TR1</a>, included a set of 23 mathematical functions that significantly |
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| 79 | * extended the standard transcendental functions inherited from C and declared |
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| 80 | * in @<cmath@>. |
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| 81 | * |
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| 82 | * Although most components from TR1 were eventually adopted for C++11 these |
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| 83 | * math functions were left behind out of concern for implementability. |
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| 84 | * The math functions were published as a separate international standard |
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| 85 | * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2010/n3060.pdf"> |
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| 86 | * IS 29124 - Extensions to the C++ Library to Support Mathematical Special |
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| 87 | * Functions</a>. |
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| 88 | * |
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| 89 | * For C++17 these functions were incorporated into the main standard. |
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| 90 | * |
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| 91 | * @subsection mathsf_contents Contents |
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| 92 | * The following functions are implemented in namespace @c std: |
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| 93 | * - @ref assoc_laguerre "assoc_laguerre - Associated Laguerre functions" |
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| 94 | * - @ref assoc_legendre "assoc_legendre - Associated Legendre functions" |
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| 95 | * - @ref beta "beta - Beta functions" |
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| 96 | * - @ref comp_ellint_1 "comp_ellint_1 - Complete elliptic functions of the first kind" |
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| 97 | * - @ref comp_ellint_2 "comp_ellint_2 - Complete elliptic functions of the second kind" |
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| 98 | * - @ref comp_ellint_3 "comp_ellint_3 - Complete elliptic functions of the third kind" |
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| 99 | * - @ref cyl_bessel_i "cyl_bessel_i - Regular modified cylindrical Bessel functions" |
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| 100 | * - @ref cyl_bessel_j "cyl_bessel_j - Cylindrical Bessel functions of the first kind" |
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| 101 | * - @ref cyl_bessel_k "cyl_bessel_k - Irregular modified cylindrical Bessel functions" |
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| 102 | * - @ref cyl_neumann "cyl_neumann - Cylindrical Neumann functions or Cylindrical Bessel functions of the second kind" |
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| 103 | * - @ref ellint_1 "ellint_1 - Incomplete elliptic functions of the first kind" |
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| 104 | * - @ref ellint_2 "ellint_2 - Incomplete elliptic functions of the second kind" |
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| 105 | * - @ref ellint_3 "ellint_3 - Incomplete elliptic functions of the third kind" |
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| 106 | * - @ref expint "expint - The exponential integral" |
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| 107 | * - @ref hermite "hermite - Hermite polynomials" |
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| 108 | * - @ref laguerre "laguerre - Laguerre functions" |
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| 109 | * - @ref legendre "legendre - Legendre polynomials" |
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| 110 | * - @ref riemann_zeta "riemann_zeta - The Riemann zeta function" |
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| 111 | * - @ref sph_bessel "sph_bessel - Spherical Bessel functions" |
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| 112 | * - @ref sph_legendre "sph_legendre - Spherical Legendre functions" |
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| 113 | * - @ref sph_neumann "sph_neumann - Spherical Neumann functions" |
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| 114 | * |
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| 115 | * The hypergeometric functions were stricken from the TR29124 and C++17 |
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| 116 | * versions of this math library because of implementation concerns. |
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| 117 | * However, since they were in the TR1 version and since they are popular |
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| 118 | * we kept them as an extension in namespace @c __gnu_cxx: |
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| 119 | * - @ref __gnu_cxx::conf_hyperg "conf_hyperg - Confluent hypergeometric functions" |
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| 120 | * - @ref __gnu_cxx::hyperg "hyperg - Hypergeometric functions" |
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| 121 | * |
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| 122 | * <!-- @subsection mathsf_general General Features --> |
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| 123 | * |
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| 124 | * @subsection mathsf_promotion Argument Promotion |
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| 125 | * The arguments suppled to the non-suffixed functions will be promoted |
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| 126 | * according to the following rules: |
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| 127 | * 1. If any argument intended to be floating point is given an integral value |
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| 128 | * That integral value is promoted to double. |
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| 129 | * 2. All floating point arguments are promoted up to the largest floating |
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| 130 | * point precision among them. |
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| 131 | * |
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| 132 | * @subsection mathsf_NaN NaN Arguments |
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| 133 | * If any of the floating point arguments supplied to these functions is |
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| 134 | * invalid or NaN (std::numeric_limits<Tp>::quiet_NaN), |
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| 135 | * the value NaN is returned. |
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| 136 | * |
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| 137 | * @subsection mathsf_impl Implementation |
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| 138 | * |
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| 139 | * We strive to implement the underlying math with type generic algorithms |
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| 140 | * to the greatest extent possible. In practice, the functions are thin |
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| 141 | * wrappers that dispatch to function templates. Type dependence is |
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| 142 | * controlled with std::numeric_limits and functions thereof. |
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| 143 | * |
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| 144 | * We don't promote @c float to @c double or @c double to <tt>long double</tt> |
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| 145 | * reflexively. The goal is for @c float functions to operate more quickly, |
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| 146 | * at the cost of @c float accuracy and possibly a smaller domain of validity. |
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| 147 | * Similaryly, <tt>long double</tt> should give you more dynamic range |
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| 148 | * and slightly more pecision than @c double on many systems. |
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| 149 | * |
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| 150 | * @subsection mathsf_testing Testing |
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| 151 | * |
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| 152 | * These functions have been tested against equivalent implementations |
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| 153 | * from the <a href="http://www.gnu.org/software/gsl"> |
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| 154 | * Gnu Scientific Library, GSL</a> and |
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| 155 | * <a href="http://www.boost.org/doc/libs/1_60_0/libs/math/doc/html/index.html">Boost</a> |
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| 156 | * and the ratio |
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| 157 | * @f[ |
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| 158 | * \frac{|f - f_{test}|}{|f_{test}|} |
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| 159 | * @f] |
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| 160 | * is generally found to be within 10<sup>-15</sup> for 64-bit double on |
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| 161 | * linux-x86_64 systems over most of the ranges of validity. |
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| 162 | * |
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| 163 | * @todo Provide accuracy comparisons on a per-function basis for a small |
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| 164 | * number of targets. |
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| 165 | * |
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| 166 | * @subsection mathsf_bibliography General Bibliography |
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| 167 | * |
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| 168 | * @see Abramowitz and Stegun: Handbook of Mathematical Functions, |
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| 169 | * with Formulas, Graphs, and Mathematical Tables |
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| 170 | * Edited by Milton Abramowitz and Irene A. Stegun, |
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| 171 | * National Bureau of Standards Applied Mathematics Series - 55 |
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| 172 | * Issued June 1964, Tenth Printing, December 1972, with corrections |
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| 173 | * Electronic versions of A&S abound including both pdf and navigable html. |
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| 174 | * @see for example http://people.math.sfu.ca/~cbm/aands/ |
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| 175 | * |
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| 176 | * @see The old A&S has been redone as the |
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| 177 | * NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/ |
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| 178 | * This version is far more navigable and includes more recent work. |
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| 179 | * |
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| 180 | * @see An Atlas of Functions: with Equator, the Atlas Function Calculator |
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| 181 | * 2nd Edition, by Oldham, Keith B., Myland, Jan, Spanier, Jerome |
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| 182 | * |
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| 183 | * @see Asymptotics and Special Functions by Frank W. J. Olver, |
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| 184 | * Academic Press, 1974 |
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| 185 | * |
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| 186 | * @see Numerical Recipes in C, The Art of Scientific Computing, |
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| 187 | * by William H. Press, Second Ed., Saul A. Teukolsky, |
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| 188 | * William T. Vetterling, and Brian P. Flannery, |
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| 189 | * Cambridge University Press, 1992 |
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| 190 | * |
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| 191 | * @see The Special Functions and Their Approximations: Volumes 1 and 2, |
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| 192 | * by Yudell L. Luke, Academic Press, 1969 |
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| 193 | * |
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| 194 | * @{ |
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| 195 | */ |
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| 196 | |
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| 197 | // Associated Laguerre polynomials |
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| 198 | |
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| 199 | /** |
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| 200 | * Return the associated Laguerre polynomial of order @c n, |
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| 201 | * degree @c m: @f$ L_n^m(x) @f$ for @c float argument. |
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| 202 | * |
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| 203 | * @see assoc_laguerre for more details. |
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| 204 | */ |
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| 205 | inline float |
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| 206 | assoc_laguerref(unsigned int __n, unsigned int __m, float __x) |
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| 207 | { return __detail::__assoc_laguerre<float>(__n, __m, __x); } |
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| 208 | |
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| 209 | /** |
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| 210 | * Return the associated Laguerre polynomial of order @c n, |
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| 211 | * degree @c m: @f$ L_n^m(x) @f$. |
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| 212 | * |
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| 213 | * @see assoc_laguerre for more details. |
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| 214 | */ |
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| 215 | inline long double |
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| 216 | assoc_laguerrel(unsigned int __n, unsigned int __m, long double __x) |
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| 217 | { return __detail::__assoc_laguerre<long double>(__n, __m, __x); } |
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| 218 | |
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| 219 | /** |
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| 220 | * Return the associated Laguerre polynomial of nonnegative order @c n, |
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| 221 | * nonnegative degree @c m and real argument @c x: @f$ L_n^m(x) @f$. |
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| 222 | * |
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| 223 | * The associated Laguerre function of real degree @f$ \alpha @f$, |
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| 224 | * @f$ L_n^\alpha(x) @f$, is defined by |
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| 225 | * @f[ |
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| 226 | * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} |
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| 227 | * {}_1F_1(-n; \alpha + 1; x) |
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| 228 | * @f] |
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| 229 | * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and |
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| 230 | * @f$ {}_1F_1(a; c; x) @f$ is the confluent hypergeometric function. |
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| 231 | * |
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| 232 | * The associated Laguerre polynomial is defined for integral |
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| 233 | * degree @f$ \alpha = m @f$ by: |
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| 234 | * @f[ |
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| 235 | * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) |
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| 236 | * @f] |
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| 237 | * where the Laguerre polynomial is defined by: |
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| 238 | * @f[ |
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| 239 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |
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| 240 | * @f] |
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| 241 | * and @f$ x >= 0 @f$. |
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| 242 | * @see laguerre for details of the Laguerre function of degree @c n |
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| 243 | * |
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| 244 | * @tparam _Tp The floating-point type of the argument @c __x. |
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| 245 | * @param __n The order of the Laguerre function, <tt>__n >= 0</tt>. |
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| 246 | * @param __m The degree of the Laguerre function, <tt>__m >= 0</tt>. |
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| 247 | * @param __x The argument of the Laguerre function, <tt>__x >= 0</tt>. |
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| 248 | * @throw std::domain_error if <tt>__x < 0</tt>. |
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| 249 | */ |
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| 250 | template<typename _Tp> |
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| 251 | inline typename __gnu_cxx::__promote<_Tp>::__type |
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| 252 | assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x) |
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| 253 | { |
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| 254 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
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| 255 | return __detail::__assoc_laguerre<__type>(__n, __m, __x); |
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| 256 | } |
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| 257 | |
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| 258 | // Associated Legendre functions |
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| 259 | |
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| 260 | /** |
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| 261 | * Return the associated Legendre function of degree @c l and order @c m |
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| 262 | * for @c float argument. |
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| 263 | * |
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| 264 | * @see assoc_legendre for more details. |
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| 265 | */ |
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| 266 | inline float |
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| 267 | assoc_legendref(unsigned int __l, unsigned int __m, float __x) |
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| 268 | { return __detail::__assoc_legendre_p<float>(__l, __m, __x); } |
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| 269 | |
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| 270 | /** |
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| 271 | * Return the associated Legendre function of degree @c l and order @c m. |
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| 272 | * |
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| 273 | * @see assoc_legendre for more details. |
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| 274 | */ |
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| 275 | inline long double |
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| 276 | assoc_legendrel(unsigned int __l, unsigned int __m, long double __x) |
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| 277 | { return __detail::__assoc_legendre_p<long double>(__l, __m, __x); } |
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| 278 | |
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| 279 | |
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| 280 | /** |
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| 281 | * Return the associated Legendre function of degree @c l and order @c m. |
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| 282 | * |
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| 283 | * The associated Legendre function is derived from the Legendre function |
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| 284 | * @f$ P_l(x) @f$ by the Rodrigues formula: |
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| 285 | * @f[ |
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| 286 | * P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x) |
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| 287 | * @f] |
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| 288 | * @see legendre for details of the Legendre function of degree @c l |
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| 289 | * |
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| 290 | * @tparam _Tp The floating-point type of the argument @c __x. |
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| 291 | * @param __l The degree <tt>__l >= 0</tt>. |
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| 292 | * @param __m The order <tt>__m <= l</tt>. |
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| 293 | * @param __x The argument, <tt>abs(__x) <= 1</tt>. |
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| 294 | * @throw std::domain_error if <tt>abs(__x) > 1</tt>. |
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| 295 | */ |
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| 296 | template<typename _Tp> |
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| 297 | inline typename __gnu_cxx::__promote<_Tp>::__type |
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| 298 | assoc_legendre(unsigned int __l, unsigned int __m, _Tp __x) |
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| 299 | { |
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| 300 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
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| 301 | return __detail::__assoc_legendre_p<__type>(__l, __m, __x); |
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| 302 | } |
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| 303 | |
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| 304 | // Beta functions |
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| 305 | |
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| 306 | /** |
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| 307 | * Return the beta function, @f$ B(a,b) @f$, for @c float parameters @c a, @c b. |
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| 308 | * |
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| 309 | * @see beta for more details. |
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| 310 | */ |
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| 311 | inline float |
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| 312 | betaf(float __a, float __b) |
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| 313 | { return __detail::__beta<float>(x: __a, y: __b); } |
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| 314 | |
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| 315 | /** |
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| 316 | * Return the beta function, @f$B(a,b)@f$, for long double |
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| 317 | * parameters @c a, @c b. |
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| 318 | * |
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| 319 | * @see beta for more details. |
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| 320 | */ |
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| 321 | inline long double |
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| 322 | betal(long double __a, long double __b) |
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| 323 | { return __detail::__beta<long double>(x: __a, y: __b); } |
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| 324 | |
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| 325 | /** |
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| 326 | * Return the beta function, @f$B(a,b)@f$, for real parameters @c a, @c b. |
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| 327 | * |
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| 328 | * The beta function is defined by |
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| 329 | * @f[ |
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| 330 | * B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt |
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| 331 | * = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} |
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| 332 | * @f] |
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| 333 | * where @f$ a > 0 @f$ and @f$ b > 0 @f$ |
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| 334 | * |
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| 335 | * @tparam _Tpa The floating-point type of the parameter @c __a. |
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| 336 | * @tparam _Tpb The floating-point type of the parameter @c __b. |
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| 337 | * @param __a The first argument of the beta function, <tt> __a > 0 </tt>. |
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| 338 | * @param __b The second argument of the beta function, <tt> __b > 0 </tt>. |
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| 339 | * @throw std::domain_error if <tt> __a < 0 </tt> or <tt> __b < 0 </tt>. |
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| 340 | */ |
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| 341 | template<typename _Tpa, typename _Tpb> |
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| 342 | inline typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type |
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| 343 | beta(_Tpa __a, _Tpb __b) |
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| 344 | { |
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| 345 | typedef typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type __type; |
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| 346 | return __detail::__beta<__type>(__a, __b); |
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| 347 | } |
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| 348 | |
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| 349 | // Complete elliptic integrals of the first kind |
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| 350 | |
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| 351 | /** |
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| 352 | * Return the complete elliptic integral of the first kind @f$ E(k) @f$ |
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| 353 | * for @c float modulus @c k. |
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| 354 | * |
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| 355 | * @see comp_ellint_1 for details. |
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| 356 | */ |
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| 357 | inline float |
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| 358 | comp_ellint_1f(float __k) |
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| 359 | { return __detail::__comp_ellint_1<float>(__k); } |
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| 360 | |
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| 361 | /** |
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| 362 | * Return the complete elliptic integral of the first kind @f$ E(k) @f$ |
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| 363 | * for long double modulus @c k. |
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| 364 | * |
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| 365 | * @see comp_ellint_1 for details. |
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| 366 | */ |
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| 367 | inline long double |
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| 368 | comp_ellint_1l(long double __k) |
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| 369 | { return __detail::__comp_ellint_1<long double>(__k); } |
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| 370 | |
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| 371 | /** |
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| 372 | * Return the complete elliptic integral of the first kind |
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| 373 | * @f$ K(k) @f$ for real modulus @c k. |
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| 374 | * |
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| 375 | * The complete elliptic integral of the first kind is defined as |
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| 376 | * @f[ |
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| 377 | * K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} |
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| 378 | * {\sqrt{1 - k^2 sin^2\theta}} |
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| 379 | * @f] |
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| 380 | * where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the |
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| 381 | * first kind and the modulus @f$ |k| <= 1 @f$. |
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| 382 | * @see ellint_1 for details of the incomplete elliptic function |
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| 383 | * of the first kind. |
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| 384 | * |
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| 385 | * @tparam _Tp The floating-point type of the modulus @c __k. |
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| 386 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> |
|---|
| 387 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. |
|---|
| 388 | */ |
|---|
| 389 | template<typename _Tp> |
|---|
| 390 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 391 | comp_ellint_1(_Tp __k) |
|---|
| 392 | { |
|---|
| 393 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 394 | return __detail::__comp_ellint_1<__type>(__k); |
|---|
| 395 | } |
|---|
| 396 | |
|---|
| 397 | // Complete elliptic integrals of the second kind |
|---|
| 398 | |
|---|
| 399 | /** |
|---|
| 400 | * Return the complete elliptic integral of the second kind @f$ E(k) @f$ |
|---|
| 401 | * for @c float modulus @c k. |
|---|
| 402 | * |
|---|
| 403 | * @see comp_ellint_2 for details. |
|---|
| 404 | */ |
|---|
| 405 | inline float |
|---|
| 406 | comp_ellint_2f(float __k) |
|---|
| 407 | { return __detail::__comp_ellint_2<float>(__k); } |
|---|
| 408 | |
|---|
| 409 | /** |
|---|
| 410 | * Return the complete elliptic integral of the second kind @f$ E(k) @f$ |
|---|
| 411 | * for long double modulus @c k. |
|---|
| 412 | * |
|---|
| 413 | * @see comp_ellint_2 for details. |
|---|
| 414 | */ |
|---|
| 415 | inline long double |
|---|
| 416 | comp_ellint_2l(long double __k) |
|---|
| 417 | { return __detail::__comp_ellint_2<long double>(__k); } |
|---|
| 418 | |
|---|
| 419 | /** |
|---|
| 420 | * Return the complete elliptic integral of the second kind @f$ E(k) @f$ |
|---|
| 421 | * for real modulus @c k. |
|---|
| 422 | * |
|---|
| 423 | * The complete elliptic integral of the second kind is defined as |
|---|
| 424 | * @f[ |
|---|
| 425 | * E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} |
|---|
| 426 | * @f] |
|---|
| 427 | * where @f$ E(k,\phi) @f$ is the incomplete elliptic integral of the |
|---|
| 428 | * second kind and the modulus @f$ |k| <= 1 @f$. |
|---|
| 429 | * @see ellint_2 for details of the incomplete elliptic function |
|---|
| 430 | * of the second kind. |
|---|
| 431 | * |
|---|
| 432 | * @tparam _Tp The floating-point type of the modulus @c __k. |
|---|
| 433 | * @param __k The modulus, @c abs(__k) <= 1 |
|---|
| 434 | * @throw std::domain_error if @c abs(__k) > 1. |
|---|
| 435 | */ |
|---|
| 436 | template<typename _Tp> |
|---|
| 437 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 438 | comp_ellint_2(_Tp __k) |
|---|
| 439 | { |
|---|
| 440 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 441 | return __detail::__comp_ellint_2<__type>(__k); |
|---|
| 442 | } |
|---|
| 443 | |
|---|
| 444 | // Complete elliptic integrals of the third kind |
|---|
| 445 | |
|---|
| 446 | /** |
|---|
| 447 | * @brief Return the complete elliptic integral of the third kind |
|---|
| 448 | * @f$ \Pi(k,\nu) @f$ for @c float modulus @c k. |
|---|
| 449 | * |
|---|
| 450 | * @see comp_ellint_3 for details. |
|---|
| 451 | */ |
|---|
| 452 | inline float |
|---|
| 453 | comp_ellint_3f(float __k, float __nu) |
|---|
| 454 | { return __detail::__comp_ellint_3<float>(__k, __nu); } |
|---|
| 455 | |
|---|
| 456 | /** |
|---|
| 457 | * @brief Return the complete elliptic integral of the third kind |
|---|
| 458 | * @f$ \Pi(k,\nu) @f$ for <tt>long double</tt> modulus @c k. |
|---|
| 459 | * |
|---|
| 460 | * @see comp_ellint_3 for details. |
|---|
| 461 | */ |
|---|
| 462 | inline long double |
|---|
| 463 | comp_ellint_3l(long double __k, long double __nu) |
|---|
| 464 | { return __detail::__comp_ellint_3<long double>(__k, __nu); } |
|---|
| 465 | |
|---|
| 466 | /** |
|---|
| 467 | * Return the complete elliptic integral of the third kind |
|---|
| 468 | * @f$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) @f$ for real modulus @c k. |
|---|
| 469 | * |
|---|
| 470 | * The complete elliptic integral of the third kind is defined as |
|---|
| 471 | * @f[ |
|---|
| 472 | * \Pi(k,\nu) = \Pi(k,\nu,\pi/2) = \int_0^{\pi/2} |
|---|
| 473 | * \frac{d\theta} |
|---|
| 474 | * {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} |
|---|
| 475 | * @f] |
|---|
| 476 | * where @f$ \Pi(k,\nu,\phi) @f$ is the incomplete elliptic integral of the |
|---|
| 477 | * second kind and the modulus @f$ |k| <= 1 @f$. |
|---|
| 478 | * @see ellint_3 for details of the incomplete elliptic function |
|---|
| 479 | * of the third kind. |
|---|
| 480 | * |
|---|
| 481 | * @tparam _Tp The floating-point type of the modulus @c __k. |
|---|
| 482 | * @tparam _Tpn The floating-point type of the argument @c __nu. |
|---|
| 483 | * @param __k The modulus, @c abs(__k) <= 1 |
|---|
| 484 | * @param __nu The argument |
|---|
| 485 | * @throw std::domain_error if @c abs(__k) > 1. |
|---|
| 486 | */ |
|---|
| 487 | template<typename _Tp, typename _Tpn> |
|---|
| 488 | inline typename __gnu_cxx::__promote_2<_Tp, _Tpn>::__type |
|---|
| 489 | comp_ellint_3(_Tp __k, _Tpn __nu) |
|---|
| 490 | { |
|---|
| 491 | typedef typename __gnu_cxx::__promote_2<_Tp, _Tpn>::__type __type; |
|---|
| 492 | return __detail::__comp_ellint_3<__type>(__k, __nu); |
|---|
| 493 | } |
|---|
| 494 | |
|---|
| 495 | // Regular modified cylindrical Bessel functions |
|---|
| 496 | |
|---|
| 497 | /** |
|---|
| 498 | * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ |
|---|
| 499 | * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
|---|
| 500 | * |
|---|
| 501 | * @see cyl_bessel_i for setails. |
|---|
| 502 | */ |
|---|
| 503 | inline float |
|---|
| 504 | cyl_bessel_if(float __nu, float __x) |
|---|
| 505 | { return __detail::__cyl_bessel_i<float>(__nu, __x); } |
|---|
| 506 | |
|---|
| 507 | /** |
|---|
| 508 | * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ |
|---|
| 509 | * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
|---|
| 510 | * |
|---|
| 511 | * @see cyl_bessel_i for setails. |
|---|
| 512 | */ |
|---|
| 513 | inline long double |
|---|
| 514 | cyl_bessel_il(long double __nu, long double __x) |
|---|
| 515 | { return __detail::__cyl_bessel_i<long double>(__nu, __x); } |
|---|
| 516 | |
|---|
| 517 | /** |
|---|
| 518 | * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ |
|---|
| 519 | * for real order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
|---|
| 520 | * |
|---|
| 521 | * The regular modified cylindrical Bessel function is: |
|---|
| 522 | * @f[ |
|---|
| 523 | * I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty} |
|---|
| 524 | * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
|---|
| 525 | * @f] |
|---|
| 526 | * |
|---|
| 527 | * @tparam _Tpnu The floating-point type of the order @c __nu. |
|---|
| 528 | * @tparam _Tp The floating-point type of the argument @c __x. |
|---|
| 529 | * @param __nu The order |
|---|
| 530 | * @param __x The argument, <tt> __x >= 0 </tt> |
|---|
| 531 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
|---|
| 532 | */ |
|---|
| 533 | template<typename _Tpnu, typename _Tp> |
|---|
| 534 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type |
|---|
| 535 | cyl_bessel_i(_Tpnu __nu, _Tp __x) |
|---|
| 536 | { |
|---|
| 537 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; |
|---|
| 538 | return __detail::__cyl_bessel_i<__type>(__nu, __x); |
|---|
| 539 | } |
|---|
| 540 | |
|---|
| 541 | // Cylindrical Bessel functions (of the first kind) |
|---|
| 542 | |
|---|
| 543 | /** |
|---|
| 544 | * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$ |
|---|
| 545 | * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
|---|
| 546 | * |
|---|
| 547 | * @see cyl_bessel_j for setails. |
|---|
| 548 | */ |
|---|
| 549 | inline float |
|---|
| 550 | cyl_bessel_jf(float __nu, float __x) |
|---|
| 551 | { return __detail::__cyl_bessel_j<float>(__nu, __x); } |
|---|
| 552 | |
|---|
| 553 | /** |
|---|
| 554 | * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$ |
|---|
| 555 | * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
|---|
| 556 | * |
|---|
| 557 | * @see cyl_bessel_j for setails. |
|---|
| 558 | */ |
|---|
| 559 | inline long double |
|---|
| 560 | cyl_bessel_jl(long double __nu, long double __x) |
|---|
| 561 | { return __detail::__cyl_bessel_j<long double>(__nu, __x); } |
|---|
| 562 | |
|---|
| 563 | /** |
|---|
| 564 | * Return the Bessel function @f$ J_{\nu}(x) @f$ of real order @f$ \nu @f$ |
|---|
| 565 | * and argument @f$ x >= 0 @f$. |
|---|
| 566 | * |
|---|
| 567 | * The cylindrical Bessel function is: |
|---|
| 568 | * @f[ |
|---|
| 569 | * J_{\nu}(x) = \sum_{k=0}^{\infty} |
|---|
| 570 | * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} |
|---|
| 571 | * @f] |
|---|
| 572 | * |
|---|
| 573 | * @tparam _Tpnu The floating-point type of the order @c __nu. |
|---|
| 574 | * @tparam _Tp The floating-point type of the argument @c __x. |
|---|
| 575 | * @param __nu The order |
|---|
| 576 | * @param __x The argument, <tt> __x >= 0 </tt> |
|---|
| 577 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
|---|
| 578 | */ |
|---|
| 579 | template<typename _Tpnu, typename _Tp> |
|---|
| 580 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type |
|---|
| 581 | cyl_bessel_j(_Tpnu __nu, _Tp __x) |
|---|
| 582 | { |
|---|
| 583 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; |
|---|
| 584 | return __detail::__cyl_bessel_j<__type>(__nu, __x); |
|---|
| 585 | } |
|---|
| 586 | |
|---|
| 587 | // Irregular modified cylindrical Bessel functions |
|---|
| 588 | |
|---|
| 589 | /** |
|---|
| 590 | * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ |
|---|
| 591 | * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
|---|
| 592 | * |
|---|
| 593 | * @see cyl_bessel_k for setails. |
|---|
| 594 | */ |
|---|
| 595 | inline float |
|---|
| 596 | cyl_bessel_kf(float __nu, float __x) |
|---|
| 597 | { return __detail::__cyl_bessel_k<float>(__nu, __x); } |
|---|
| 598 | |
|---|
| 599 | /** |
|---|
| 600 | * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ |
|---|
| 601 | * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
|---|
| 602 | * |
|---|
| 603 | * @see cyl_bessel_k for setails. |
|---|
| 604 | */ |
|---|
| 605 | inline long double |
|---|
| 606 | cyl_bessel_kl(long double __nu, long double __x) |
|---|
| 607 | { return __detail::__cyl_bessel_k<long double>(__nu, __x); } |
|---|
| 608 | |
|---|
| 609 | /** |
|---|
| 610 | * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ |
|---|
| 611 | * of real order @f$ \nu @f$ and argument @f$ x @f$. |
|---|
| 612 | * |
|---|
| 613 | * The irregular modified Bessel function is defined by: |
|---|
| 614 | * @f[ |
|---|
| 615 | * K_{\nu}(x) = \frac{\pi}{2} |
|---|
| 616 | * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} |
|---|
| 617 | * @f] |
|---|
| 618 | * where for integral @f$ \nu = n @f$ a limit is taken: |
|---|
| 619 | * @f$ lim_{\nu \to n} @f$. |
|---|
| 620 | * For negative argument we have simply: |
|---|
| 621 | * @f[ |
|---|
| 622 | * K_{-\nu}(x) = K_{\nu}(x) |
|---|
| 623 | * @f] |
|---|
| 624 | * |
|---|
| 625 | * @tparam _Tpnu The floating-point type of the order @c __nu. |
|---|
| 626 | * @tparam _Tp The floating-point type of the argument @c __x. |
|---|
| 627 | * @param __nu The order |
|---|
| 628 | * @param __x The argument, <tt> __x >= 0 </tt> |
|---|
| 629 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
|---|
| 630 | */ |
|---|
| 631 | template<typename _Tpnu, typename _Tp> |
|---|
| 632 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type |
|---|
| 633 | cyl_bessel_k(_Tpnu __nu, _Tp __x) |
|---|
| 634 | { |
|---|
| 635 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; |
|---|
| 636 | return __detail::__cyl_bessel_k<__type>(__nu, __x); |
|---|
| 637 | } |
|---|
| 638 | |
|---|
| 639 | // Cylindrical Neumann functions |
|---|
| 640 | |
|---|
| 641 | /** |
|---|
| 642 | * Return the Neumann function @f$ N_{\nu}(x) @f$ |
|---|
| 643 | * of @c float order @f$ \nu @f$ and argument @f$ x @f$. |
|---|
| 644 | * |
|---|
| 645 | * @see cyl_neumann for setails. |
|---|
| 646 | */ |
|---|
| 647 | inline float |
|---|
| 648 | cyl_neumannf(float __nu, float __x) |
|---|
| 649 | { return __detail::__cyl_neumann_n<float>(__nu, __x); } |
|---|
| 650 | |
|---|
| 651 | /** |
|---|
| 652 | * Return the Neumann function @f$ N_{\nu}(x) @f$ |
|---|
| 653 | * of <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x @f$. |
|---|
| 654 | * |
|---|
| 655 | * @see cyl_neumann for setails. |
|---|
| 656 | */ |
|---|
| 657 | inline long double |
|---|
| 658 | cyl_neumannl(long double __nu, long double __x) |
|---|
| 659 | { return __detail::__cyl_neumann_n<long double>(__nu, __x); } |
|---|
| 660 | |
|---|
| 661 | /** |
|---|
| 662 | * Return the Neumann function @f$ N_{\nu}(x) @f$ |
|---|
| 663 | * of real order @f$ \nu @f$ and argument @f$ x >= 0 @f$. |
|---|
| 664 | * |
|---|
| 665 | * The Neumann function is defined by: |
|---|
| 666 | * @f[ |
|---|
| 667 | * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} |
|---|
| 668 | * {\sin \nu\pi} |
|---|
| 669 | * @f] |
|---|
| 670 | * where @f$ x >= 0 @f$ and for integral order @f$ \nu = n @f$ |
|---|
| 671 | * a limit is taken: @f$ lim_{\nu \to n} @f$. |
|---|
| 672 | * |
|---|
| 673 | * @tparam _Tpnu The floating-point type of the order @c __nu. |
|---|
| 674 | * @tparam _Tp The floating-point type of the argument @c __x. |
|---|
| 675 | * @param __nu The order |
|---|
| 676 | * @param __x The argument, <tt> __x >= 0 </tt> |
|---|
| 677 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
|---|
| 678 | */ |
|---|
| 679 | template<typename _Tpnu, typename _Tp> |
|---|
| 680 | inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type |
|---|
| 681 | cyl_neumann(_Tpnu __nu, _Tp __x) |
|---|
| 682 | { |
|---|
| 683 | typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; |
|---|
| 684 | return __detail::__cyl_neumann_n<__type>(__nu, __x); |
|---|
| 685 | } |
|---|
| 686 | |
|---|
| 687 | // Incomplete elliptic integrals of the first kind |
|---|
| 688 | |
|---|
| 689 | /** |
|---|
| 690 | * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$ |
|---|
| 691 | * for @c float modulus @f$ k @f$ and angle @f$ \phi @f$. |
|---|
| 692 | * |
|---|
| 693 | * @see ellint_1 for details. |
|---|
| 694 | */ |
|---|
| 695 | inline float |
|---|
| 696 | ellint_1f(float __k, float __phi) |
|---|
| 697 | { return __detail::__ellint_1<float>(__k, __phi); } |
|---|
| 698 | |
|---|
| 699 | /** |
|---|
| 700 | * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$ |
|---|
| 701 | * for <tt>long double</tt> modulus @f$ k @f$ and angle @f$ \phi @f$. |
|---|
| 702 | * |
|---|
| 703 | * @see ellint_1 for details. |
|---|
| 704 | */ |
|---|
| 705 | inline long double |
|---|
| 706 | ellint_1l(long double __k, long double __phi) |
|---|
| 707 | { return __detail::__ellint_1<long double>(__k, __phi); } |
|---|
| 708 | |
|---|
| 709 | /** |
|---|
| 710 | * Return the incomplete elliptic integral of the first kind @f$ F(k,\phi) @f$ |
|---|
| 711 | * for @c real modulus @f$ k @f$ and angle @f$ \phi @f$. |
|---|
| 712 | * |
|---|
| 713 | * The incomplete elliptic integral of the first kind is defined as |
|---|
| 714 | * @f[ |
|---|
| 715 | * F(k,\phi) = \int_0^{\phi}\frac{d\theta} |
|---|
| 716 | * {\sqrt{1 - k^2 sin^2\theta}} |
|---|
| 717 | * @f] |
|---|
| 718 | * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of |
|---|
| 719 | * the first kind, @f$ K(k) @f$. @see comp_ellint_1. |
|---|
| 720 | * |
|---|
| 721 | * @tparam _Tp The floating-point type of the modulus @c __k. |
|---|
| 722 | * @tparam _Tpp The floating-point type of the angle @c __phi. |
|---|
| 723 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> |
|---|
| 724 | * @param __phi The integral limit argument in radians |
|---|
| 725 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. |
|---|
| 726 | */ |
|---|
| 727 | template<typename _Tp, typename _Tpp> |
|---|
| 728 | inline typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type |
|---|
| 729 | ellint_1(_Tp __k, _Tpp __phi) |
|---|
| 730 | { |
|---|
| 731 | typedef typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type __type; |
|---|
| 732 | return __detail::__ellint_1<__type>(__k, __phi); |
|---|
| 733 | } |
|---|
| 734 | |
|---|
| 735 | // Incomplete elliptic integrals of the second kind |
|---|
| 736 | |
|---|
| 737 | /** |
|---|
| 738 | * @brief Return the incomplete elliptic integral of the second kind |
|---|
| 739 | * @f$ E(k,\phi) @f$ for @c float argument. |
|---|
| 740 | * |
|---|
| 741 | * @see ellint_2 for details. |
|---|
| 742 | */ |
|---|
| 743 | inline float |
|---|
| 744 | ellint_2f(float __k, float __phi) |
|---|
| 745 | { return __detail::__ellint_2<float>(__k, __phi); } |
|---|
| 746 | |
|---|
| 747 | /** |
|---|
| 748 | * @brief Return the incomplete elliptic integral of the second kind |
|---|
| 749 | * @f$ E(k,\phi) @f$. |
|---|
| 750 | * |
|---|
| 751 | * @see ellint_2 for details. |
|---|
| 752 | */ |
|---|
| 753 | inline long double |
|---|
| 754 | ellint_2l(long double __k, long double __phi) |
|---|
| 755 | { return __detail::__ellint_2<long double>(__k, __phi); } |
|---|
| 756 | |
|---|
| 757 | /** |
|---|
| 758 | * Return the incomplete elliptic integral of the second kind |
|---|
| 759 | * @f$ E(k,\phi) @f$. |
|---|
| 760 | * |
|---|
| 761 | * The incomplete elliptic integral of the second kind is defined as |
|---|
| 762 | * @f[ |
|---|
| 763 | * E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} |
|---|
| 764 | * @f] |
|---|
| 765 | * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of |
|---|
| 766 | * the second kind, @f$ E(k) @f$. @see comp_ellint_2. |
|---|
| 767 | * |
|---|
| 768 | * @tparam _Tp The floating-point type of the modulus @c __k. |
|---|
| 769 | * @tparam _Tpp The floating-point type of the angle @c __phi. |
|---|
| 770 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> |
|---|
| 771 | * @param __phi The integral limit argument in radians |
|---|
| 772 | * @return The elliptic function of the second kind. |
|---|
| 773 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. |
|---|
| 774 | */ |
|---|
| 775 | template<typename _Tp, typename _Tpp> |
|---|
| 776 | inline typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type |
|---|
| 777 | ellint_2(_Tp __k, _Tpp __phi) |
|---|
| 778 | { |
|---|
| 779 | typedef typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type __type; |
|---|
| 780 | return __detail::__ellint_2<__type>(__k, __phi); |
|---|
| 781 | } |
|---|
| 782 | |
|---|
| 783 | // Incomplete elliptic integrals of the third kind |
|---|
| 784 | |
|---|
| 785 | /** |
|---|
| 786 | * @brief Return the incomplete elliptic integral of the third kind |
|---|
| 787 | * @f$ \Pi(k,\nu,\phi) @f$ for @c float argument. |
|---|
| 788 | * |
|---|
| 789 | * @see ellint_3 for details. |
|---|
| 790 | */ |
|---|
| 791 | inline float |
|---|
| 792 | ellint_3f(float __k, float __nu, float __phi) |
|---|
| 793 | { return __detail::__ellint_3<float>(__k, __nu, __phi); } |
|---|
| 794 | |
|---|
| 795 | /** |
|---|
| 796 | * @brief Return the incomplete elliptic integral of the third kind |
|---|
| 797 | * @f$ \Pi(k,\nu,\phi) @f$. |
|---|
| 798 | * |
|---|
| 799 | * @see ellint_3 for details. |
|---|
| 800 | */ |
|---|
| 801 | inline long double |
|---|
| 802 | ellint_3l(long double __k, long double __nu, long double __phi) |
|---|
| 803 | { return __detail::__ellint_3<long double>(__k, __nu, __phi); } |
|---|
| 804 | |
|---|
| 805 | /** |
|---|
| 806 | * @brief Return the incomplete elliptic integral of the third kind |
|---|
| 807 | * @f$ \Pi(k,\nu,\phi) @f$. |
|---|
| 808 | * |
|---|
| 809 | * The incomplete elliptic integral of the third kind is defined by: |
|---|
| 810 | * @f[ |
|---|
| 811 | * \Pi(k,\nu,\phi) = \int_0^{\phi} |
|---|
| 812 | * \frac{d\theta} |
|---|
| 813 | * {(1 - \nu \sin^2\theta) |
|---|
| 814 | * \sqrt{1 - k^2 \sin^2\theta}} |
|---|
| 815 | * @f] |
|---|
| 816 | * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of |
|---|
| 817 | * the third kind, @f$ \Pi(k,\nu) @f$. @see comp_ellint_3. |
|---|
| 818 | * |
|---|
| 819 | * @tparam _Tp The floating-point type of the modulus @c __k. |
|---|
| 820 | * @tparam _Tpn The floating-point type of the argument @c __nu. |
|---|
| 821 | * @tparam _Tpp The floating-point type of the angle @c __phi. |
|---|
| 822 | * @param __k The modulus, <tt> abs(__k) <= 1 </tt> |
|---|
| 823 | * @param __nu The second argument |
|---|
| 824 | * @param __phi The integral limit argument in radians |
|---|
| 825 | * @return The elliptic function of the third kind. |
|---|
| 826 | * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. |
|---|
| 827 | */ |
|---|
| 828 | template<typename _Tp, typename _Tpn, typename _Tpp> |
|---|
| 829 | inline typename __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type |
|---|
| 830 | ellint_3(_Tp __k, _Tpn __nu, _Tpp __phi) |
|---|
| 831 | { |
|---|
| 832 | typedef typename __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type __type; |
|---|
| 833 | return __detail::__ellint_3<__type>(__k, __nu, __phi); |
|---|
| 834 | } |
|---|
| 835 | |
|---|
| 836 | // Exponential integrals |
|---|
| 837 | |
|---|
| 838 | /** |
|---|
| 839 | * Return the exponential integral @f$ Ei(x) @f$ for @c float argument @c x. |
|---|
| 840 | * |
|---|
| 841 | * @see expint for details. |
|---|
| 842 | */ |
|---|
| 843 | inline float |
|---|
| 844 | expintf(float __x) |
|---|
| 845 | { return __detail::__expint<float>(__x); } |
|---|
| 846 | |
|---|
| 847 | /** |
|---|
| 848 | * Return the exponential integral @f$ Ei(x) @f$ |
|---|
| 849 | * for <tt>long double</tt> argument @c x. |
|---|
| 850 | * |
|---|
| 851 | * @see expint for details. |
|---|
| 852 | */ |
|---|
| 853 | inline long double |
|---|
| 854 | expintl(long double __x) |
|---|
| 855 | { return __detail::__expint<long double>(__x); } |
|---|
| 856 | |
|---|
| 857 | /** |
|---|
| 858 | * Return the exponential integral @f$ Ei(x) @f$ for @c real argument @c x. |
|---|
| 859 | * |
|---|
| 860 | * The exponential integral is given by |
|---|
| 861 | * \f[ |
|---|
| 862 | * Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt |
|---|
| 863 | * \f] |
|---|
| 864 | * |
|---|
| 865 | * @tparam _Tp The floating-point type of the argument @c __x. |
|---|
| 866 | * @param __x The argument of the exponential integral function. |
|---|
| 867 | */ |
|---|
| 868 | template<typename _Tp> |
|---|
| 869 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 870 | expint(_Tp __x) |
|---|
| 871 | { |
|---|
| 872 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 873 | return __detail::__expint<__type>(__x); |
|---|
| 874 | } |
|---|
| 875 | |
|---|
| 876 | // Hermite polynomials |
|---|
| 877 | |
|---|
| 878 | /** |
|---|
| 879 | * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n |
|---|
| 880 | * and float argument @c x. |
|---|
| 881 | * |
|---|
| 882 | * @see hermite for details. |
|---|
| 883 | */ |
|---|
| 884 | inline float |
|---|
| 885 | hermitef(unsigned int __n, float __x) |
|---|
| 886 | { return __detail::__poly_hermite<float>(__n, __x); } |
|---|
| 887 | |
|---|
| 888 | /** |
|---|
| 889 | * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n |
|---|
| 890 | * and <tt>long double</tt> argument @c x. |
|---|
| 891 | * |
|---|
| 892 | * @see hermite for details. |
|---|
| 893 | */ |
|---|
| 894 | inline long double |
|---|
| 895 | hermitel(unsigned int __n, long double __x) |
|---|
| 896 | { return __detail::__poly_hermite<long double>(__n, __x); } |
|---|
| 897 | |
|---|
| 898 | /** |
|---|
| 899 | * Return the Hermite polynomial @f$ H_n(x) @f$ of order n |
|---|
| 900 | * and @c real argument @c x. |
|---|
| 901 | * |
|---|
| 902 | * The Hermite polynomial is defined by: |
|---|
| 903 | * @f[ |
|---|
| 904 | * H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} |
|---|
| 905 | * @f] |
|---|
| 906 | * |
|---|
| 907 | * The Hermite polynomial obeys a reflection formula: |
|---|
| 908 | * @f[ |
|---|
| 909 | * H_n(-x) = (-1)^n H_n(x) |
|---|
| 910 | * @f] |
|---|
| 911 | * |
|---|
| 912 | * @tparam _Tp The floating-point type of the argument @c __x. |
|---|
| 913 | * @param __n The order |
|---|
| 914 | * @param __x The argument |
|---|
| 915 | */ |
|---|
| 916 | template<typename _Tp> |
|---|
| 917 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 918 | hermite(unsigned int __n, _Tp __x) |
|---|
| 919 | { |
|---|
| 920 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 921 | return __detail::__poly_hermite<__type>(__n, __x); |
|---|
| 922 | } |
|---|
| 923 | |
|---|
| 924 | // Laguerre polynomials |
|---|
| 925 | |
|---|
| 926 | /** |
|---|
| 927 | * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n |
|---|
| 928 | * and @c float argument @f$ x >= 0 @f$. |
|---|
| 929 | * |
|---|
| 930 | * @see laguerre for more details. |
|---|
| 931 | */ |
|---|
| 932 | inline float |
|---|
| 933 | laguerref(unsigned int __n, float __x) |
|---|
| 934 | { return __detail::__laguerre<float>(__n, __x); } |
|---|
| 935 | |
|---|
| 936 | /** |
|---|
| 937 | * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n |
|---|
| 938 | * and <tt>long double</tt> argument @f$ x >= 0 @f$. |
|---|
| 939 | * |
|---|
| 940 | * @see laguerre for more details. |
|---|
| 941 | */ |
|---|
| 942 | inline long double |
|---|
| 943 | laguerrel(unsigned int __n, long double __x) |
|---|
| 944 | { return __detail::__laguerre<long double>(__n, __x); } |
|---|
| 945 | |
|---|
| 946 | /** |
|---|
| 947 | * Returns the Laguerre polynomial @f$ L_n(x) @f$ |
|---|
| 948 | * of nonnegative degree @c n and real argument @f$ x >= 0 @f$. |
|---|
| 949 | * |
|---|
| 950 | * The Laguerre polynomial is defined by: |
|---|
| 951 | * @f[ |
|---|
| 952 | * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) |
|---|
| 953 | * @f] |
|---|
| 954 | * |
|---|
| 955 | * @tparam _Tp The floating-point type of the argument @c __x. |
|---|
| 956 | * @param __n The nonnegative order |
|---|
| 957 | * @param __x The argument <tt> __x >= 0 </tt> |
|---|
| 958 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
|---|
| 959 | */ |
|---|
| 960 | template<typename _Tp> |
|---|
| 961 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 962 | laguerre(unsigned int __n, _Tp __x) |
|---|
| 963 | { |
|---|
| 964 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 965 | return __detail::__laguerre<__type>(__n, __x); |
|---|
| 966 | } |
|---|
| 967 | |
|---|
| 968 | // Legendre polynomials |
|---|
| 969 | |
|---|
| 970 | /** |
|---|
| 971 | * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative |
|---|
| 972 | * degree @f$ l @f$ and @c float argument @f$ |x| <= 0 @f$. |
|---|
| 973 | * |
|---|
| 974 | * @see legendre for more details. |
|---|
| 975 | */ |
|---|
| 976 | inline float |
|---|
| 977 | legendref(unsigned int __l, float __x) |
|---|
| 978 | { return __detail::__poly_legendre_p<float>(__l, __x); } |
|---|
| 979 | |
|---|
| 980 | /** |
|---|
| 981 | * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative |
|---|
| 982 | * degree @f$ l @f$ and <tt>long double</tt> argument @f$ |x| <= 0 @f$. |
|---|
| 983 | * |
|---|
| 984 | * @see legendre for more details. |
|---|
| 985 | */ |
|---|
| 986 | inline long double |
|---|
| 987 | legendrel(unsigned int __l, long double __x) |
|---|
| 988 | { return __detail::__poly_legendre_p<long double>(__l, __x); } |
|---|
| 989 | |
|---|
| 990 | /** |
|---|
| 991 | * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative |
|---|
| 992 | * degree @f$ l @f$ and real argument @f$ |x| <= 0 @f$. |
|---|
| 993 | * |
|---|
| 994 | * The Legendre function of order @f$ l @f$ and argument @f$ x @f$, |
|---|
| 995 | * @f$ P_l(x) @f$, is defined by: |
|---|
| 996 | * @f[ |
|---|
| 997 | * P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l} |
|---|
| 998 | * @f] |
|---|
| 999 | * |
|---|
| 1000 | * @tparam _Tp The floating-point type of the argument @c __x. |
|---|
| 1001 | * @param __l The degree @f$ l >= 0 @f$ |
|---|
| 1002 | * @param __x The argument @c abs(__x) <= 1 |
|---|
| 1003 | * @throw std::domain_error if @c abs(__x) > 1 |
|---|
| 1004 | */ |
|---|
| 1005 | template<typename _Tp> |
|---|
| 1006 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 1007 | legendre(unsigned int __l, _Tp __x) |
|---|
| 1008 | { |
|---|
| 1009 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 1010 | return __detail::__poly_legendre_p<__type>(__l, __x); |
|---|
| 1011 | } |
|---|
| 1012 | |
|---|
| 1013 | // Riemann zeta functions |
|---|
| 1014 | |
|---|
| 1015 | /** |
|---|
| 1016 | * Return the Riemann zeta function @f$ \zeta(s) @f$ |
|---|
| 1017 | * for @c float argument @f$ s @f$. |
|---|
| 1018 | * |
|---|
| 1019 | * @see riemann_zeta for more details. |
|---|
| 1020 | */ |
|---|
| 1021 | inline float |
|---|
| 1022 | riemann_zetaf(float __s) |
|---|
| 1023 | { return __detail::__riemann_zeta<float>(__s); } |
|---|
| 1024 | |
|---|
| 1025 | /** |
|---|
| 1026 | * Return the Riemann zeta function @f$ \zeta(s) @f$ |
|---|
| 1027 | * for <tt>long double</tt> argument @f$ s @f$. |
|---|
| 1028 | * |
|---|
| 1029 | * @see riemann_zeta for more details. |
|---|
| 1030 | */ |
|---|
| 1031 | inline long double |
|---|
| 1032 | riemann_zetal(long double __s) |
|---|
| 1033 | { return __detail::__riemann_zeta<long double>(__s); } |
|---|
| 1034 | |
|---|
| 1035 | /** |
|---|
| 1036 | * Return the Riemann zeta function @f$ \zeta(s) @f$ |
|---|
| 1037 | * for real argument @f$ s @f$. |
|---|
| 1038 | * |
|---|
| 1039 | * The Riemann zeta function is defined by: |
|---|
| 1040 | * @f[ |
|---|
| 1041 | * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1 |
|---|
| 1042 | * @f] |
|---|
| 1043 | * and |
|---|
| 1044 | * @f[ |
|---|
| 1045 | * \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s} |
|---|
| 1046 | * \hbox{ for } 0 <= s <= 1 |
|---|
| 1047 | * @f] |
|---|
| 1048 | * For s < 1 use the reflection formula: |
|---|
| 1049 | * @f[ |
|---|
| 1050 | * \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s) |
|---|
| 1051 | * @f] |
|---|
| 1052 | * |
|---|
| 1053 | * @tparam _Tp The floating-point type of the argument @c __s. |
|---|
| 1054 | * @param __s The argument <tt> s != 1 </tt> |
|---|
| 1055 | */ |
|---|
| 1056 | template<typename _Tp> |
|---|
| 1057 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 1058 | riemann_zeta(_Tp __s) |
|---|
| 1059 | { |
|---|
| 1060 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 1061 | return __detail::__riemann_zeta<__type>(__s); |
|---|
| 1062 | } |
|---|
| 1063 | |
|---|
| 1064 | // Spherical Bessel functions |
|---|
| 1065 | |
|---|
| 1066 | /** |
|---|
| 1067 | * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n |
|---|
| 1068 | * and @c float argument @f$ x >= 0 @f$. |
|---|
| 1069 | * |
|---|
| 1070 | * @see sph_bessel for more details. |
|---|
| 1071 | */ |
|---|
| 1072 | inline float |
|---|
| 1073 | sph_besself(unsigned int __n, float __x) |
|---|
| 1074 | { return __detail::__sph_bessel<float>(__n, __x); } |
|---|
| 1075 | |
|---|
| 1076 | /** |
|---|
| 1077 | * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n |
|---|
| 1078 | * and <tt>long double</tt> argument @f$ x >= 0 @f$. |
|---|
| 1079 | * |
|---|
| 1080 | * @see sph_bessel for more details. |
|---|
| 1081 | */ |
|---|
| 1082 | inline long double |
|---|
| 1083 | sph_bessell(unsigned int __n, long double __x) |
|---|
| 1084 | { return __detail::__sph_bessel<long double>(__n, __x); } |
|---|
| 1085 | |
|---|
| 1086 | /** |
|---|
| 1087 | * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n |
|---|
| 1088 | * and real argument @f$ x >= 0 @f$. |
|---|
| 1089 | * |
|---|
| 1090 | * The spherical Bessel function is defined by: |
|---|
| 1091 | * @f[ |
|---|
| 1092 | * j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) |
|---|
| 1093 | * @f] |
|---|
| 1094 | * |
|---|
| 1095 | * @tparam _Tp The floating-point type of the argument @c __x. |
|---|
| 1096 | * @param __n The integral order <tt> n >= 0 </tt> |
|---|
| 1097 | * @param __x The real argument <tt> x >= 0 </tt> |
|---|
| 1098 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
|---|
| 1099 | */ |
|---|
| 1100 | template<typename _Tp> |
|---|
| 1101 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 1102 | sph_bessel(unsigned int __n, _Tp __x) |
|---|
| 1103 | { |
|---|
| 1104 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 1105 | return __detail::__sph_bessel<__type>(__n, __x); |
|---|
| 1106 | } |
|---|
| 1107 | |
|---|
| 1108 | // Spherical associated Legendre functions |
|---|
| 1109 | |
|---|
| 1110 | /** |
|---|
| 1111 | * Return the spherical Legendre function of nonnegative integral |
|---|
| 1112 | * degree @c l and order @c m and float angle @f$ \theta @f$ in radians. |
|---|
| 1113 | * |
|---|
| 1114 | * @see sph_legendre for details. |
|---|
| 1115 | */ |
|---|
| 1116 | inline float |
|---|
| 1117 | sph_legendref(unsigned int __l, unsigned int __m, float __theta) |
|---|
| 1118 | { return __detail::__sph_legendre<float>(__l, __m, __theta); } |
|---|
| 1119 | |
|---|
| 1120 | /** |
|---|
| 1121 | * Return the spherical Legendre function of nonnegative integral |
|---|
| 1122 | * degree @c l and order @c m and <tt>long double</tt> angle @f$ \theta @f$ |
|---|
| 1123 | * in radians. |
|---|
| 1124 | * |
|---|
| 1125 | * @see sph_legendre for details. |
|---|
| 1126 | */ |
|---|
| 1127 | inline long double |
|---|
| 1128 | sph_legendrel(unsigned int __l, unsigned int __m, long double __theta) |
|---|
| 1129 | { return __detail::__sph_legendre<long double>(__l, __m, __theta); } |
|---|
| 1130 | |
|---|
| 1131 | /** |
|---|
| 1132 | * Return the spherical Legendre function of nonnegative integral |
|---|
| 1133 | * degree @c l and order @c m and real angle @f$ \theta @f$ in radians. |
|---|
| 1134 | * |
|---|
| 1135 | * The spherical Legendre function is defined by |
|---|
| 1136 | * @f[ |
|---|
| 1137 | * Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi} |
|---|
| 1138 | * \frac{(l-m)!}{(l+m)!}] |
|---|
| 1139 | * P_l^m(\cos\theta) \exp^{im\phi} |
|---|
| 1140 | * @f] |
|---|
| 1141 | * |
|---|
| 1142 | * @tparam _Tp The floating-point type of the angle @c __theta. |
|---|
| 1143 | * @param __l The order <tt> __l >= 0 </tt> |
|---|
| 1144 | * @param __m The degree <tt> __m >= 0 </tt> and <tt> __m <= __l </tt> |
|---|
| 1145 | * @param __theta The radian polar angle argument |
|---|
| 1146 | */ |
|---|
| 1147 | template<typename _Tp> |
|---|
| 1148 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 1149 | sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta) |
|---|
| 1150 | { |
|---|
| 1151 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 1152 | return __detail::__sph_legendre<__type>(__l, __m, __theta); |
|---|
| 1153 | } |
|---|
| 1154 | |
|---|
| 1155 | // Spherical Neumann functions |
|---|
| 1156 | |
|---|
| 1157 | /** |
|---|
| 1158 | * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ |
|---|
| 1159 | * and @c float argument @f$ x >= 0 @f$. |
|---|
| 1160 | * |
|---|
| 1161 | * @see sph_neumann for details. |
|---|
| 1162 | */ |
|---|
| 1163 | inline float |
|---|
| 1164 | sph_neumannf(unsigned int __n, float __x) |
|---|
| 1165 | { return __detail::__sph_neumann<float>(__n, __x); } |
|---|
| 1166 | |
|---|
| 1167 | /** |
|---|
| 1168 | * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ |
|---|
| 1169 | * and <tt>long double</tt> @f$ x >= 0 @f$. |
|---|
| 1170 | * |
|---|
| 1171 | * @see sph_neumann for details. |
|---|
| 1172 | */ |
|---|
| 1173 | inline long double |
|---|
| 1174 | sph_neumannl(unsigned int __n, long double __x) |
|---|
| 1175 | { return __detail::__sph_neumann<long double>(__n, __x); } |
|---|
| 1176 | |
|---|
| 1177 | /** |
|---|
| 1178 | * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ |
|---|
| 1179 | * and real argument @f$ x >= 0 @f$. |
|---|
| 1180 | * |
|---|
| 1181 | * The spherical Neumann function is defined by |
|---|
| 1182 | * @f[ |
|---|
| 1183 | * n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) |
|---|
| 1184 | * @f] |
|---|
| 1185 | * |
|---|
| 1186 | * @tparam _Tp The floating-point type of the argument @c __x. |
|---|
| 1187 | * @param __n The integral order <tt> n >= 0 </tt> |
|---|
| 1188 | * @param __x The real argument <tt> __x >= 0 </tt> |
|---|
| 1189 | * @throw std::domain_error if <tt> __x < 0 </tt>. |
|---|
| 1190 | */ |
|---|
| 1191 | template<typename _Tp> |
|---|
| 1192 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 1193 | sph_neumann(unsigned int __n, _Tp __x) |
|---|
| 1194 | { |
|---|
| 1195 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 1196 | return __detail::__sph_neumann<__type>(__n, __x); |
|---|
| 1197 | } |
|---|
| 1198 | |
|---|
| 1199 | /// @} group mathsf |
|---|
| 1200 | |
|---|
| 1201 | _GLIBCXX_END_NAMESPACE_VERSION |
|---|
| 1202 | } // namespace std |
|---|
| 1203 | |
|---|
| 1204 | #ifndef __STRICT_ANSI__ |
|---|
| 1205 | namespace __gnu_cxx _GLIBCXX_VISIBILITY(default) |
|---|
| 1206 | { |
|---|
| 1207 | _GLIBCXX_BEGIN_NAMESPACE_VERSION |
|---|
| 1208 | |
|---|
| 1209 | /** @addtogroup mathsf |
|---|
| 1210 | * @{ |
|---|
| 1211 | */ |
|---|
| 1212 | |
|---|
| 1213 | // Airy functions |
|---|
| 1214 | |
|---|
| 1215 | /** |
|---|
| 1216 | * Return the Airy function @f$ Ai(x) @f$ of @c float argument x. |
|---|
| 1217 | */ |
|---|
| 1218 | inline float |
|---|
| 1219 | airy_aif(float __x) |
|---|
| 1220 | { |
|---|
| 1221 | float __Ai, __Bi, __Aip, __Bip; |
|---|
| 1222 | std::__detail::__airy<float>(__x, __Ai, __Bi, __Aip, __Bip); |
|---|
| 1223 | return __Ai; |
|---|
| 1224 | } |
|---|
| 1225 | |
|---|
| 1226 | /** |
|---|
| 1227 | * Return the Airy function @f$ Ai(x) @f$ of <tt>long double</tt> argument x. |
|---|
| 1228 | */ |
|---|
| 1229 | inline long double |
|---|
| 1230 | airy_ail(long double __x) |
|---|
| 1231 | { |
|---|
| 1232 | long double __Ai, __Bi, __Aip, __Bip; |
|---|
| 1233 | std::__detail::__airy<long double>(__x, __Ai, __Bi, __Aip, __Bip); |
|---|
| 1234 | return __Ai; |
|---|
| 1235 | } |
|---|
| 1236 | |
|---|
| 1237 | /** |
|---|
| 1238 | * Return the Airy function @f$ Ai(x) @f$ of real argument x. |
|---|
| 1239 | */ |
|---|
| 1240 | template<typename _Tp> |
|---|
| 1241 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 1242 | airy_ai(_Tp __x) |
|---|
| 1243 | { |
|---|
| 1244 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 1245 | __type __Ai, __Bi, __Aip, __Bip; |
|---|
| 1246 | std::__detail::__airy<__type>(__x, __Ai, __Bi, __Aip, __Bip); |
|---|
| 1247 | return __Ai; |
|---|
| 1248 | } |
|---|
| 1249 | |
|---|
| 1250 | /** |
|---|
| 1251 | * Return the Airy function @f$ Bi(x) @f$ of @c float argument x. |
|---|
| 1252 | */ |
|---|
| 1253 | inline float |
|---|
| 1254 | airy_bif(float __x) |
|---|
| 1255 | { |
|---|
| 1256 | float __Ai, __Bi, __Aip, __Bip; |
|---|
| 1257 | std::__detail::__airy<float>(__x, __Ai, __Bi, __Aip, __Bip); |
|---|
| 1258 | return __Bi; |
|---|
| 1259 | } |
|---|
| 1260 | |
|---|
| 1261 | /** |
|---|
| 1262 | * Return the Airy function @f$ Bi(x) @f$ of <tt>long double</tt> argument x. |
|---|
| 1263 | */ |
|---|
| 1264 | inline long double |
|---|
| 1265 | airy_bil(long double __x) |
|---|
| 1266 | { |
|---|
| 1267 | long double __Ai, __Bi, __Aip, __Bip; |
|---|
| 1268 | std::__detail::__airy<long double>(__x, __Ai, __Bi, __Aip, __Bip); |
|---|
| 1269 | return __Bi; |
|---|
| 1270 | } |
|---|
| 1271 | |
|---|
| 1272 | /** |
|---|
| 1273 | * Return the Airy function @f$ Bi(x) @f$ of real argument x. |
|---|
| 1274 | */ |
|---|
| 1275 | template<typename _Tp> |
|---|
| 1276 | inline typename __gnu_cxx::__promote<_Tp>::__type |
|---|
| 1277 | airy_bi(_Tp __x) |
|---|
| 1278 | { |
|---|
| 1279 | typedef typename __gnu_cxx::__promote<_Tp>::__type __type; |
|---|
| 1280 | __type __Ai, __Bi, __Aip, __Bip; |
|---|
| 1281 | std::__detail::__airy<__type>(__x, __Ai, __Bi, __Aip, __Bip); |
|---|
| 1282 | return __Bi; |
|---|
| 1283 | } |
|---|
| 1284 | |
|---|
| 1285 | // Confluent hypergeometric functions |
|---|
| 1286 | |
|---|
| 1287 | /** |
|---|
| 1288 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ |
|---|
| 1289 | * of @c float numeratorial parameter @c a, denominatorial parameter @c c, |
|---|
| 1290 | * and argument @c x. |
|---|
| 1291 | * |
|---|
| 1292 | * @see conf_hyperg for details. |
|---|
| 1293 | */ |
|---|
| 1294 | inline float |
|---|
| 1295 | conf_hypergf(float __a, float __c, float __x) |
|---|
| 1296 | { return std::__detail::__conf_hyperg<float>(__a, __c, __x); } |
|---|
| 1297 | |
|---|
| 1298 | /** |
|---|
| 1299 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ |
|---|
| 1300 | * of <tt>long double</tt> numeratorial parameter @c a, |
|---|
| 1301 | * denominatorial parameter @c c, and argument @c x. |
|---|
| 1302 | * |
|---|
| 1303 | * @see conf_hyperg for details. |
|---|
| 1304 | */ |
|---|
| 1305 | inline long double |
|---|
| 1306 | conf_hypergl(long double __a, long double __c, long double __x) |
|---|
| 1307 | { return std::__detail::__conf_hyperg<long double>(__a, __c, __x); } |
|---|
| 1308 | |
|---|
| 1309 | /** |
|---|
| 1310 | * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ |
|---|
| 1311 | * of real numeratorial parameter @c a, denominatorial parameter @c c, |
|---|
| 1312 | * and argument @c x. |
|---|
| 1313 | * |
|---|
| 1314 | * The confluent hypergeometric function is defined by |
|---|
| 1315 | * @f[ |
|---|
| 1316 | * {}_1F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n x^n}{(c)_n n!} |
|---|
| 1317 | * @f] |
|---|
| 1318 | * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$, |
|---|
| 1319 | * @f$ (x)_0 = 1 @f$ |
|---|
| 1320 | * |
|---|
| 1321 | * @param __a The numeratorial parameter |
|---|
| 1322 | * @param __c The denominatorial parameter |
|---|
| 1323 | * @param __x The argument |
|---|
| 1324 | */ |
|---|
| 1325 | template<typename _Tpa, typename _Tpc, typename _Tp> |
|---|
| 1326 | inline typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type |
|---|
| 1327 | conf_hyperg(_Tpa __a, _Tpc __c, _Tp __x) |
|---|
| 1328 | { |
|---|
| 1329 | typedef typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type __type; |
|---|
| 1330 | return std::__detail::__conf_hyperg<__type>(__a, __c, __x); |
|---|
| 1331 | } |
|---|
| 1332 | |
|---|
| 1333 | // Hypergeometric functions |
|---|
| 1334 | |
|---|
| 1335 | /** |
|---|
| 1336 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ |
|---|
| 1337 | * of @ float numeratorial parameters @c a and @c b, |
|---|
| 1338 | * denominatorial parameter @c c, and argument @c x. |
|---|
| 1339 | * |
|---|
| 1340 | * @see hyperg for details. |
|---|
| 1341 | */ |
|---|
| 1342 | inline float |
|---|
| 1343 | hypergf(float __a, float __b, float __c, float __x) |
|---|
| 1344 | { return std::__detail::__hyperg<float>(__a, __b, __c, __x); } |
|---|
| 1345 | |
|---|
| 1346 | /** |
|---|
| 1347 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ |
|---|
| 1348 | * of <tt>long double</tt> numeratorial parameters @c a and @c b, |
|---|
| 1349 | * denominatorial parameter @c c, and argument @c x. |
|---|
| 1350 | * |
|---|
| 1351 | * @see hyperg for details. |
|---|
| 1352 | */ |
|---|
| 1353 | inline long double |
|---|
| 1354 | hypergl(long double __a, long double __b, long double __c, long double __x) |
|---|
| 1355 | { return std::__detail::__hyperg<long double>(__a, __b, __c, __x); } |
|---|
| 1356 | |
|---|
| 1357 | /** |
|---|
| 1358 | * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ |
|---|
| 1359 | * of real numeratorial parameters @c a and @c b, |
|---|
| 1360 | * denominatorial parameter @c c, and argument @c x. |
|---|
| 1361 | * |
|---|
| 1362 | * The hypergeometric function is defined by |
|---|
| 1363 | * @f[ |
|---|
| 1364 | * {}_2F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n x^n}{(c)_n n!} |
|---|
| 1365 | * @f] |
|---|
| 1366 | * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$, |
|---|
| 1367 | * @f$ (x)_0 = 1 @f$ |
|---|
| 1368 | * |
|---|
| 1369 | * @param __a The first numeratorial parameter |
|---|
| 1370 | * @param __b The second numeratorial parameter |
|---|
| 1371 | * @param __c The denominatorial parameter |
|---|
| 1372 | * @param __x The argument |
|---|
| 1373 | */ |
|---|
| 1374 | template<typename _Tpa, typename _Tpb, typename _Tpc, typename _Tp> |
|---|
| 1375 | inline typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp>::__type |
|---|
| 1376 | hyperg(_Tpa __a, _Tpb __b, _Tpc __c, _Tp __x) |
|---|
| 1377 | { |
|---|
| 1378 | typedef typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp> |
|---|
| 1379 | ::__type __type; |
|---|
| 1380 | return std::__detail::__hyperg<__type>(__a, __b, __c, __x); |
|---|
| 1381 | } |
|---|
| 1382 | |
|---|
| 1383 | /// @} |
|---|
| 1384 | _GLIBCXX_END_NAMESPACE_VERSION |
|---|
| 1385 | } // namespace __gnu_cxx |
|---|
| 1386 | #endif // __STRICT_ANSI__ |
|---|
| 1387 | |
|---|
| 1388 | #pragma GCC visibility pop |
|---|
| 1389 | |
|---|
| 1390 | #endif // _GLIBCXX_BITS_SPECFUN_H |
|---|
| 1391 | |
|---|
