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<a href="a00512.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno"> 1</span> <span class="comment">// Mathematical Special Functions for -*- C++ -*-</span></div><div class="line"><a name="l00002"></a><span class="lineno"> 2</span> </div><div class="line"><a name="l00003"></a><span class="lineno"> 3</span> <span class="comment">// Copyright (C) 2006-2018 Free Software Foundation, Inc.</span></div><div class="line"><a name="l00004"></a><span class="lineno"> 4</span> <span class="comment">//</span></div><div class="line"><a name="l00005"></a><span class="lineno"> 5</span> <span class="comment">// This file is part of the GNU ISO C++ Library. This library is free</span></div><div class="line"><a name="l00006"></a><span class="lineno"> 6</span> <span class="comment">// software; you can redistribute it and/or modify it under the</span></div><div class="line"><a name="l00007"></a><span class="lineno"> 7</span> <span class="comment">// terms of the GNU General Public License as published by the</span></div><div class="line"><a name="l00008"></a><span class="lineno"> 8</span> <span class="comment">// Free Software Foundation; either version 3, or (at your option)</span></div><div class="line"><a name="l00009"></a><span class="lineno"> 9</span> <span class="comment">// any later version.</span></div><div class="line"><a name="l00010"></a><span class="lineno"> 10</span> </div><div class="line"><a name="l00011"></a><span class="lineno"> 11</span> <span class="comment">// This library is distributed in the hope that it will be useful,</span></div><div class="line"><a name="l00012"></a><span class="lineno"> 12</span> <span class="comment">// but WITHOUT ANY WARRANTY; without even the implied warranty of</span></div><div class="line"><a name="l00013"></a><span class="lineno"> 13</span> <span class="comment">// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the</span></div><div class="line"><a name="l00014"></a><span class="lineno"> 14</span> <span class="comment">// GNU General Public License for more details.</span></div><div class="line"><a name="l00015"></a><span class="lineno"> 15</span> </div><div class="line"><a name="l00016"></a><span class="lineno"> 16</span> <span class="comment">// Under Section 7 of GPL version 3, you are granted additional</span></div><div class="line"><a name="l00017"></a><span class="lineno"> 17</span> <span class="comment">// permissions described in the GCC Runtime Library Exception, version</span></div><div class="line"><a name="l00018"></a><span class="lineno"> 18</span> <span class="comment">// 3.1, as published by the Free Software Foundation.</span></div><div class="line"><a name="l00019"></a><span class="lineno"> 19</span> </div><div class="line"><a name="l00020"></a><span class="lineno"> 20</span> <span class="comment">// You should have received a copy of the GNU General Public License and</span></div><div class="line"><a name="l00021"></a><span class="lineno"> 21</span> <span class="comment">// a copy of the GCC Runtime Library Exception along with this program;</span></div><div class="line"><a name="l00022"></a><span class="lineno"> 22</span> <span class="comment">// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see</span></div><div class="line"><a name="l00023"></a><span class="lineno"> 23</span> <span class="comment">// <http://www.gnu.org/licenses/>.</span></div><div class="line"><a name="l00024"></a><span class="lineno"> 24</span> <span class="comment"></span></div><div class="line"><a name="l00025"></a><span class="lineno"> 25</span> <span class="comment">/** @file bits/specfun.h</span></div><div class="line"><a name="l00026"></a><span class="lineno"> 26</span> <span class="comment"> * This is an internal header file, included by other library headers.</span></div><div class="line"><a name="l00027"></a><span class="lineno"> 27</span> <span class="comment"> * Do not attempt to use it directly. @headername{cmath}</span></div><div class="line"><a name="l00028"></a><span class="lineno"> 28</span> <span class="comment"> */</span></div><div class="line"><a name="l00029"></a><span class="lineno"> 29</span> </div><div class="line"><a name="l00030"></a><span class="lineno"> 30</span> <span class="preprocessor">#ifndef _GLIBCXX_BITS_SPECFUN_H</span></div><div class="line"><a name="l00031"></a><span class="lineno"> 31</span> <span class="preprocessor">#define _GLIBCXX_BITS_SPECFUN_H 1</span></div><div class="line"><a name="l00032"></a><span class="lineno"> 32</span> </div><div class="line"><a name="l00033"></a><span class="lineno"> 33</span> <span class="preprocessor">#pragma GCC visibility push(default)</span></div><div class="line"><a name="l00034"></a><span class="lineno"> 34</span> </div><div class="line"><a name="l00035"></a><span class="lineno"> 35</span> <span class="preprocessor">#include <<a class="code" href="a00656.html">bits/c++config.h</a>></span></div><div class="line"><a name="l00036"></a><span class="lineno"> 36</span> </div><div class="line"><a name="l00037"></a><span class="lineno"> 37</span> <span class="preprocessor">#define __STDCPP_MATH_SPEC_FUNCS__ 201003L</span></div><div class="line"><a name="l00038"></a><span class="lineno"> 38</span> </div><div class="line"><a name="l00039"></a><span class="lineno"> 39</span> <span class="preprocessor">#define __cpp_lib_math_special_functions 201603L</span></div><div class="line"><a name="l00040"></a><span class="lineno"> 40</span> </div><div class="line"><a name="l00041"></a><span class="lineno"> 41</span> <span class="preprocessor">#if __cplusplus <= 201403L && __STDCPP_WANT_MATH_SPEC_FUNCS__ == 0</span></div><div class="line"><a name="l00042"></a><span class="lineno"> 42</span> <span class="preprocessor"># error include <cmath> and define __STDCPP_WANT_MATH_SPEC_FUNCS__</span></div><div class="line"><a name="l00043"></a><span class="lineno"> 43</span> <span class="preprocessor">#endif</span></div><div class="line"><a name="l00044"></a><span class="lineno"> 44</span> </div><div class="line"><a name="l00045"></a><span class="lineno"> 45</span> <span class="preprocessor">#include <<a class="code" href="a00530.html">bits/stl_algobase.h</a>></span></div><div class="line"><a name="l00046"></a><span class="lineno"> 46</span> <span class="preprocessor">#include <<a class="code" href="a00095.html">limits</a>></span></div><div class="line"><a name="l00047"></a><span class="lineno"> 47</span> <span class="preprocessor">#include <<a class="code" href="a00167.html">type_traits</a>></span></div><div class="line"><a name="l00048"></a><span class="lineno"> 48</span> </div><div class="line"><a name="l00049"></a><span class="lineno"> 49</span> <span class="preprocessor">#include <tr1/gamma.tcc></span></div><div class="line"><a name="l00050"></a><span class="lineno"> 50</span> <span class="preprocessor">#include <tr1/bessel_function.tcc></span></div><div class="line"><a name="l00051"></a><span class="lineno"> 51</span> <span class="preprocessor">#include <tr1/beta_function.tcc></span></div><div class="line"><a name="l00052"></a><span class="lineno"> 52</span> <span class="preprocessor">#include <tr1/ell_integral.tcc></span></div><div class="line"><a name="l00053"></a><span class="lineno"> 53</span> <span class="preprocessor">#include <tr1/exp_integral.tcc></span></div><div class="line"><a name="l00054"></a><span class="lineno"> 54</span> <span class="preprocessor">#include <tr1/hypergeometric.tcc></span></div><div class="line"><a name="l00055"></a><span class="lineno"> 55</span> <span class="preprocessor">#include <tr1/legendre_function.tcc></span></div><div class="line"><a name="l00056"></a><span class="lineno"> 56</span> <span class="preprocessor">#include <tr1/modified_bessel_func.tcc></span></div><div class="line"><a name="l00057"></a><span class="lineno"> 57</span> <span class="preprocessor">#include <tr1/poly_hermite.tcc></span></div><div class="line"><a name="l00058"></a><span class="lineno"> 58</span> <span class="preprocessor">#include <tr1/poly_laguerre.tcc></span></div><div class="line"><a name="l00059"></a><span class="lineno"> 59</span> <span class="preprocessor">#include <tr1/riemann_zeta.tcc></span></div><div class="line"><a name="l00060"></a><span class="lineno"> 60</span> </div><div class="line"><a name="l00061"></a><span class="lineno"> 61</span> <span class="keyword">namespace </span><a class="code" href="a01541.html">std</a> _GLIBCXX_VISIBILITY(default)</div><div class="line"><a name="l00062"></a><span class="lineno"> 62</span> {</div><div class="line"><a name="l00063"></a><span class="lineno"> 63</span> _GLIBCXX_BEGIN_NAMESPACE_VERSION</div><div class="line"><a name="l00064"></a><span class="lineno"> 64</span> <span class="comment"></span></div><div class="line"><a name="l00065"></a><span class="lineno"> 65</span> <span class="comment"> /**</span></div><div class="line"><a name="l00066"></a><span class="lineno"> 66</span> <span class="comment"> * @defgroup mathsf Mathematical Special Functions</span></div><div class="line"><a name="l00067"></a><span class="lineno"> 67</span> <span class="comment"> * @ingroup numerics</span></div><div class="line"><a name="l00068"></a><span class="lineno"> 68</span> <span class="comment"> *</span></div><div class="line"><a name="l00069"></a><span class="lineno"> 69</span> <span class="comment"> * A collection of advanced mathematical special functions,</span></div><div class="line"><a name="l00070"></a><span class="lineno"> 70</span> <span class="comment"> * defined by ISO/IEC IS 29124.</span></div><div class="line"><a name="l00071"></a><span class="lineno"> 71</span> <span class="comment"> * @{</span></div><div class="line"><a name="l00072"></a><span class="lineno"> 72</span> <span class="comment"> */</span></div><div class="line"><a name="l00073"></a><span class="lineno"> 73</span> <span class="comment"></span></div><div class="line"><a name="l00074"></a><span class="lineno"> 74</span> <span class="comment"> /**</span></div><div class="line"><a name="l00075"></a><span class="lineno"> 75</span> <span class="comment"> * @mainpage Mathematical Special Functions</span></div><div class="line"><a name="l00076"></a><span class="lineno"> 76</span> <span class="comment"> *</span></div><div class="line"><a name="l00077"></a><span class="lineno"> 77</span> <span class="comment"> * @section intro Introduction and History</span></div><div class="line"><a name="l00078"></a><span class="lineno"> 78</span> <span class="comment"> * The first significant library upgrade on the road to C++2011,</span></div><div class="line"><a name="l00079"></a><span class="lineno"> 79</span> <span class="comment"> * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2005/n1836.pdf"></span></div><div class="line"><a name="l00080"></a><span class="lineno"> 80</span> <span class="comment"> * TR1</a>, included a set of 23 mathematical functions that significantly</span></div><div class="line"><a name="l00081"></a><span class="lineno"> 81</span> <span class="comment"> * extended the standard transcendental functions inherited from C and declared</span></div><div class="line"><a name="l00082"></a><span class="lineno"> 82</span> <span class="comment"> * in @<cmath@>.</span></div><div class="line"><a name="l00083"></a><span class="lineno"> 83</span> <span class="comment"> *</span></div><div class="line"><a name="l00084"></a><span class="lineno"> 84</span> <span class="comment"> * Although most components from TR1 were eventually adopted for C++11 these</span></div><div class="line"><a name="l00085"></a><span class="lineno"> 85</span> <span class="comment"> * math functions were left behind out of concern for implementability.</span></div><div class="line"><a name="l00086"></a><span class="lineno"> 86</span> <span class="comment"> * The math functions were published as a separate international standard</span></div><div class="line"><a name="l00087"></a><span class="lineno"> 87</span> <span class="comment"> * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2010/n3060.pdf"></span></div><div class="line"><a name="l00088"></a><span class="lineno"> 88</span> <span class="comment"> * IS 29124 - Extensions to the C++ Library to Support Mathematical Special</span></div><div class="line"><a name="l00089"></a><span class="lineno"> 89</span> <span class="comment"> * Functions</a>.</span></div><div class="line"><a name="l00090"></a><span class="lineno"> 90</span> <span class="comment"> *</span></div><div class="line"><a name="l00091"></a><span class="lineno"> 91</span> <span class="comment"> * For C++17 these functions were incorporated into the main standard.</span></div><div class="line"><a name="l00092"></a><span class="lineno"> 92</span> <span class="comment"> *</span></div><div class="line"><a name="l00093"></a><span class="lineno"> 93</span> <span class="comment"> * @section contents Contents</span></div><div class="line"><a name="l00094"></a><span class="lineno"> 94</span> <span class="comment"> * The following functions are implemented in namespace @c std:</span></div><div class="line"><a name="l00095"></a><span class="lineno"> 95</span> <span class="comment"> * - @ref assoc_laguerre "assoc_laguerre - Associated Laguerre functions"</span></div><div class="line"><a name="l00096"></a><span class="lineno"> 96</span> <span class="comment"> * - @ref assoc_legendre "assoc_legendre - Associated Legendre functions"</span></div><div class="line"><a name="l00097"></a><span class="lineno"> 97</span> <span class="comment"> * - @ref beta "beta - Beta functions"</span></div><div class="line"><a name="l00098"></a><span class="lineno"> 98</span> <span class="comment"> * - @ref comp_ellint_1 "comp_ellint_1 - Complete elliptic functions of the first kind"</span></div><div class="line"><a name="l00099"></a><span class="lineno"> 99</span> <span class="comment"> * - @ref comp_ellint_2 "comp_ellint_2 - Complete elliptic functions of the second kind"</span></div><div class="line"><a name="l00100"></a><span class="lineno"> 100</span> <span class="comment"> * - @ref comp_ellint_3 "comp_ellint_3 - Complete elliptic functions of the third kind"</span></div><div class="line"><a name="l00101"></a><span class="lineno"> 101</span> <span class="comment"> * - @ref cyl_bessel_i "cyl_bessel_i - Regular modified cylindrical Bessel functions"</span></div><div class="line"><a name="l00102"></a><span class="lineno"> 102</span> <span class="comment"> * - @ref cyl_bessel_j "cyl_bessel_j - Cylindrical Bessel functions of the first kind"</span></div><div class="line"><a name="l00103"></a><span class="lineno"> 103</span> <span class="comment"> * - @ref cyl_bessel_k "cyl_bessel_k - Irregular modified cylindrical Bessel functions"</span></div><div class="line"><a name="l00104"></a><span class="lineno"> 104</span> <span class="comment"> * - @ref cyl_neumann "cyl_neumann - Cylindrical Neumann functions or Cylindrical Bessel functions of the second kind"</span></div><div class="line"><a name="l00105"></a><span class="lineno"> 105</span> <span class="comment"> * - @ref ellint_1 "ellint_1 - Incomplete elliptic functions of the first kind"</span></div><div class="line"><a name="l00106"></a><span class="lineno"> 106</span> <span class="comment"> * - @ref ellint_2 "ellint_2 - Incomplete elliptic functions of the second kind"</span></div><div class="line"><a name="l00107"></a><span class="lineno"> 107</span> <span class="comment"> * - @ref ellint_3 "ellint_3 - Incomplete elliptic functions of the third kind"</span></div><div class="line"><a name="l00108"></a><span class="lineno"> 108</span> <span class="comment"> * - @ref expint "expint - The exponential integral"</span></div><div class="line"><a name="l00109"></a><span class="lineno"> 109</span> <span class="comment"> * - @ref hermite "hermite - Hermite polynomials"</span></div><div class="line"><a name="l00110"></a><span class="lineno"> 110</span> <span class="comment"> * - @ref laguerre "laguerre - Laguerre functions"</span></div><div class="line"><a name="l00111"></a><span class="lineno"> 111</span> <span class="comment"> * - @ref legendre "legendre - Legendre polynomials"</span></div><div class="line"><a name="l00112"></a><span class="lineno"> 112</span> <span class="comment"> * - @ref riemann_zeta "riemann_zeta - The Riemann zeta function"</span></div><div class="line"><a name="l00113"></a><span class="lineno"> 113</span> <span class="comment"> * - @ref sph_bessel "sph_bessel - Spherical Bessel functions"</span></div><div class="line"><a name="l00114"></a><span class="lineno"> 114</span> <span class="comment"> * - @ref sph_legendre "sph_legendre - Spherical Legendre functions"</span></div><div class="line"><a name="l00115"></a><span class="lineno"> 115</span> <span class="comment"> * - @ref sph_neumann "sph_neumann - Spherical Neumann functions"</span></div><div class="line"><a name="l00116"></a><span class="lineno"> 116</span> <span class="comment"> *</span></div><div class="line"><a name="l00117"></a><span class="lineno"> 117</span> <span class="comment"> * The hypergeometric functions were stricken from the TR29124 and C++17</span></div><div class="line"><a name="l00118"></a><span class="lineno"> 118</span> <span class="comment"> * versions of this math library because of implementation concerns.</span></div><div class="line"><a name="l00119"></a><span class="lineno"> 119</span> <span class="comment"> * However, since they were in the TR1 version and since they are popular</span></div><div class="line"><a name="l00120"></a><span class="lineno"> 120</span> <span class="comment"> * we kept them as an extension in namespace @c __gnu_cxx:</span></div><div class="line"><a name="l00121"></a><span class="lineno"> 121</span> <span class="comment"> * - @ref __gnu_cxx::conf_hyperg "conf_hyperg - Confluent hypergeometric functions"</span></div><div class="line"><a name="l00122"></a><span class="lineno"> 122</span> <span class="comment"> * - @ref __gnu_cxx::hyperg "hyperg - Hypergeometric functions"</span></div><div class="line"><a name="l00123"></a><span class="lineno"> 123</span> <span class="comment"> *</span></div><div class="line"><a name="l00124"></a><span class="lineno"> 124</span> <span class="comment"> * @section general General Features</span></div><div class="line"><a name="l00125"></a><span class="lineno"> 125</span> <span class="comment"> *</span></div><div class="line"><a name="l00126"></a><span class="lineno"> 126</span> <span class="comment"> * @subsection promotion Argument Promotion</span></div><div class="line"><a name="l00127"></a><span class="lineno"> 127</span> <span class="comment"> * The arguments suppled to the non-suffixed functions will be promoted</span></div><div class="line"><a name="l00128"></a><span class="lineno"> 128</span> <span class="comment"> * according to the following rules:</span></div><div class="line"><a name="l00129"></a><span class="lineno"> 129</span> <span class="comment"> * 1. If any argument intended to be floating point is given an integral value</span></div><div class="line"><a name="l00130"></a><span class="lineno"> 130</span> <span class="comment"> * That integral value is promoted to double.</span></div><div class="line"><a name="l00131"></a><span class="lineno"> 131</span> <span class="comment"> * 2. All floating point arguments are promoted up to the largest floating</span></div><div class="line"><a name="l00132"></a><span class="lineno"> 132</span> <span class="comment"> * point precision among them.</span></div><div class="line"><a name="l00133"></a><span class="lineno"> 133</span> <span class="comment"> *</span></div><div class="line"><a name="l00134"></a><span class="lineno"> 134</span> <span class="comment"> * @subsection NaN NaN Arguments</span></div><div class="line"><a name="l00135"></a><span class="lineno"> 135</span> <span class="comment"> * If any of the floating point arguments supplied to these functions is</span></div><div class="line"><a name="l00136"></a><span class="lineno"> 136</span> <span class="comment"> * invalid or NaN (std::numeric_limits<Tp>::quiet_NaN),</span></div><div class="line"><a name="l00137"></a><span class="lineno"> 137</span> <span class="comment"> * the value NaN is returned.</span></div><div class="line"><a name="l00138"></a><span class="lineno"> 138</span> <span class="comment"> *</span></div><div class="line"><a name="l00139"></a><span class="lineno"> 139</span> <span class="comment"> * @section impl Implementation</span></div><div class="line"><a name="l00140"></a><span class="lineno"> 140</span> <span class="comment"> *</span></div><div class="line"><a name="l00141"></a><span class="lineno"> 141</span> <span class="comment"> * We strive to implement the underlying math with type generic algorithms</span></div><div class="line"><a name="l00142"></a><span class="lineno"> 142</span> <span class="comment"> * to the greatest extent possible. In practice, the functions are thin</span></div><div class="line"><a name="l00143"></a><span class="lineno"> 143</span> <span class="comment"> * wrappers that dispatch to function templates. Type dependence is</span></div><div class="line"><a name="l00144"></a><span class="lineno"> 144</span> <span class="comment"> * controlled with std::numeric_limits and functions thereof.</span></div><div class="line"><a name="l00145"></a><span class="lineno"> 145</span> <span class="comment"> *</span></div><div class="line"><a name="l00146"></a><span class="lineno"> 146</span> <span class="comment"> * We don't promote @c float to @c double or @c double to <tt>long double</tt></span></div><div class="line"><a name="l00147"></a><span class="lineno"> 147</span> <span class="comment"> * reflexively. The goal is for @c float functions to operate more quickly,</span></div><div class="line"><a name="l00148"></a><span class="lineno"> 148</span> <span class="comment"> * at the cost of @c float accuracy and possibly a smaller domain of validity.</span></div><div class="line"><a name="l00149"></a><span class="lineno"> 149</span> <span class="comment"> * Similaryly, <tt>long double</tt> should give you more dynamic range</span></div><div class="line"><a name="l00150"></a><span class="lineno"> 150</span> <span class="comment"> * and slightly more pecision than @c double on many systems.</span></div><div class="line"><a name="l00151"></a><span class="lineno"> 151</span> <span class="comment"> *</span></div><div class="line"><a name="l00152"></a><span class="lineno"> 152</span> <span class="comment"> * @section testing Testing</span></div><div class="line"><a name="l00153"></a><span class="lineno"> 153</span> <span class="comment"> *</span></div><div class="line"><a name="l00154"></a><span class="lineno"> 154</span> <span class="comment"> * These functions have been tested against equivalent implementations</span></div><div class="line"><a name="l00155"></a><span class="lineno"> 155</span> <span class="comment"> * from the <a href="http://www.gnu.org/software/gsl"></span></div><div class="line"><a name="l00156"></a><span class="lineno"> 156</span> <span class="comment"> * Gnu Scientific Library, GSL</a> and</span></div><div class="line"><a name="l00157"></a><span class="lineno"> 157</span> <span class="comment"> * <a href="http://www.boost.org/doc/libs/1_60_0/libs/math/doc/html/index.html>Boost</a></span></div><div class="line"><a name="l00158"></a><span class="lineno"> 158</span> <span class="comment"> * and the ratio</span></div><div class="line"><a name="l00159"></a><span class="lineno"> 159</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00160"></a><span class="lineno"> 160</span> <span class="comment"> * \frac{|f - f_{test}|}{|f_{test}|}</span></div><div class="line"><a name="l00161"></a><span class="lineno"> 161</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00162"></a><span class="lineno"> 162</span> <span class="comment"> * is generally found to be within 10^-15 for 64-bit double on linux-x86_64 systems</span></div><div class="line"><a name="l00163"></a><span class="lineno"> 163</span> <span class="comment"> * over most of the ranges of validity.</span></div><div class="line"><a name="l00164"></a><span class="lineno"> 164</span> <span class="comment"> * </span></div><div class="line"><a name="l00165"></a><span class="lineno"> 165</span> <span class="comment"> * @todo Provide accuracy comparisons on a per-function basis for a small</span></div><div class="line"><a name="l00166"></a><span class="lineno"> 166</span> <span class="comment"> * number of targets.</span></div><div class="line"><a name="l00167"></a><span class="lineno"> 167</span> <span class="comment"> *</span></div><div class="line"><a name="l00168"></a><span class="lineno"> 168</span> <span class="comment"> * @section bibliography General Bibliography</span></div><div class="line"><a name="l00169"></a><span class="lineno"> 169</span> <span class="comment"> *</span></div><div class="line"><a name="l00170"></a><span class="lineno"> 170</span> <span class="comment"> * @see Abramowitz and Stegun: Handbook of Mathematical Functions,</span></div><div class="line"><a name="l00171"></a><span class="lineno"> 171</span> <span class="comment"> * with Formulas, Graphs, and Mathematical Tables</span></div><div class="line"><a name="l00172"></a><span class="lineno"> 172</span> <span class="comment"> * Edited by Milton Abramowitz and Irene A. Stegun,</span></div><div class="line"><a name="l00173"></a><span class="lineno"> 173</span> <span class="comment"> * National Bureau of Standards Applied Mathematics Series - 55</span></div><div class="line"><a name="l00174"></a><span class="lineno"> 174</span> <span class="comment"> * Issued June 1964, Tenth Printing, December 1972, with corrections</span></div><div class="line"><a name="l00175"></a><span class="lineno"> 175</span> <span class="comment"> * Electronic versions of A&S abound including both pdf and navigable html.</span></div><div class="line"><a name="l00176"></a><span class="lineno"> 176</span> <span class="comment"> * @see for example http://people.math.sfu.ca/~cbm/aands/</span></div><div class="line"><a name="l00177"></a><span class="lineno"> 177</span> <span class="comment"> *</span></div><div class="line"><a name="l00178"></a><span class="lineno"> 178</span> <span class="comment"> * @see The old A&S has been redone as the</span></div><div class="line"><a name="l00179"></a><span class="lineno"> 179</span> <span class="comment"> * NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/</span></div><div class="line"><a name="l00180"></a><span class="lineno"> 180</span> <span class="comment"> * This version is far more navigable and includes more recent work.</span></div><div class="line"><a name="l00181"></a><span class="lineno"> 181</span> <span class="comment"> *</span></div><div class="line"><a name="l00182"></a><span class="lineno"> 182</span> <span class="comment"> * @see An Atlas of Functions: with Equator, the Atlas Function Calculator</span></div><div class="line"><a name="l00183"></a><span class="lineno"> 183</span> <span class="comment"> * 2nd Edition, by Oldham, Keith B., Myland, Jan, Spanier, Jerome</span></div><div class="line"><a name="l00184"></a><span class="lineno"> 184</span> <span class="comment"> *</span></div><div class="line"><a name="l00185"></a><span class="lineno"> 185</span> <span class="comment"> * @see Asymptotics and Special Functions by Frank W. J. Olver,</span></div><div class="line"><a name="l00186"></a><span class="lineno"> 186</span> <span class="comment"> * Academic Press, 1974</span></div><div class="line"><a name="l00187"></a><span class="lineno"> 187</span> <span class="comment"> *</span></div><div class="line"><a name="l00188"></a><span class="lineno"> 188</span> <span class="comment"> * @see Numerical Recipes in C, The Art of Scientific Computing,</span></div><div class="line"><a name="l00189"></a><span class="lineno"> 189</span> <span class="comment"> * by William H. Press, Second Ed., Saul A. Teukolsky,</span></div><div class="line"><a name="l00190"></a><span class="lineno"> 190</span> <span class="comment"> * William T. Vetterling, and Brian P. Flannery,</span></div><div class="line"><a name="l00191"></a><span class="lineno"> 191</span> <span class="comment"> * Cambridge University Press, 1992</span></div><div class="line"><a name="l00192"></a><span class="lineno"> 192</span> <span class="comment"> *</span></div><div class="line"><a name="l00193"></a><span class="lineno"> 193</span> <span class="comment"> * @see The Special Functions and Their Approximations: Volumes 1 and 2,</span></div><div class="line"><a name="l00194"></a><span class="lineno"> 194</span> <span class="comment"> * by Yudell L. Luke, Academic Press, 1969</span></div><div class="line"><a name="l00195"></a><span class="lineno"> 195</span> <span class="comment"> */</span></div><div class="line"><a name="l00196"></a><span class="lineno"> 196</span> </div><div class="line"><a name="l00197"></a><span class="lineno"> 197</span>  <span class="comment">// Associated Laguerre polynomials</span></div><div class="line"><a name="l00198"></a><span class="lineno"> 198</span> <span class="comment"></span></div><div class="line"><a name="l00199"></a><span class="lineno"> 199</span> <span class="comment"> /**</span></div><div class="line"><a name="l00200"></a><span class="lineno"> 200</span> <span class="comment"> * Return the associated Laguerre polynomial of order @c n,</span></div><div class="line"><a name="l00201"></a><span class="lineno"> 201</span> <span class="comment"> * degree @c m: @f$ L_n^m(x) @f$ for @c float argument.</span></div><div class="line"><a name="l00202"></a><span class="lineno"> 202</span> <span class="comment"> *</span></div><div class="line"><a name="l00203"></a><span class="lineno"> 203</span> <span class="comment"> * @see assoc_laguerre for more details.</span></div><div class="line"><a name="l00204"></a><span class="lineno"> 204</span> <span class="comment"> */</span></div><div class="line"><a name="l00205"></a><span class="lineno"> 205</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00206"></a><span class="lineno"><a class="line" href="a01497.html#gaf83d98f350a1cfcebee6a1f723cf90d2"> 206</a></span>  <a class="code" href="a01497.html#gaf83d98f350a1cfcebee6a1f723cf90d2">assoc_laguerref</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __m, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l00207"></a><span class="lineno"> 207</span>  { <span class="keywordflow">return</span> __detail::__assoc_laguerre<float>(__n, __m, __x); }</div><div class="line"><a name="l00208"></a><span class="lineno"> 208</span> <span class="comment"></span></div><div class="line"><a name="l00209"></a><span class="lineno"> 209</span> <span class="comment"> /**</span></div><div class="line"><a name="l00210"></a><span class="lineno"> 210</span> <span class="comment"> * Return the associated Laguerre polynomial of order @c n,</span></div><div class="line"><a name="l00211"></a><span class="lineno"> 211</span> <span class="comment"> * degree @c m: @f$ L_n^m(x) @f$.</span></div><div class="line"><a name="l00212"></a><span class="lineno"> 212</span> <span class="comment"> *</span></div><div class="line"><a name="l00213"></a><span class="lineno"> 213</span> <span class="comment"> * @see assoc_laguerre for more details.</span></div><div class="line"><a name="l00214"></a><span class="lineno"> 214</span> <span class="comment"> */</span></div><div class="line"><a name="l00215"></a><span class="lineno"> 215</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00216"></a><span class="lineno"><a class="line" href="a01497.html#gac8e245671fb2df5de5fd978d03081f6c"> 216</a></span>  <a class="code" href="a01497.html#gac8e245671fb2df5de5fd978d03081f6c">assoc_laguerrel</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __m, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l00217"></a><span class="lineno"> 217</span>  { <span class="keywordflow">return</span> __detail::__assoc_laguerre<long double>(__n, __m, __x); }</div><div class="line"><a name="l00218"></a><span class="lineno"> 218</span> <span class="comment"></span></div><div class="line"><a name="l00219"></a><span class="lineno"> 219</span> <span class="comment"> /**</span></div><div class="line"><a name="l00220"></a><span class="lineno"> 220</span> <span class="comment"> * Return the associated Laguerre polynomial of nonnegative order @c n,</span></div><div class="line"><a name="l00221"></a><span class="lineno"> 221</span> <span class="comment"> * nonnegative degree @c m and real argument @c x: @f$ L_n^m(x) @f$.</span></div><div class="line"><a name="l00222"></a><span class="lineno"> 222</span> <span class="comment"> *</span></div><div class="line"><a name="l00223"></a><span class="lineno"> 223</span> <span class="comment"> * The associated Laguerre function of real degree @f$ \alpha @f$,</span></div><div class="line"><a name="l00224"></a><span class="lineno"> 224</span> <span class="comment"> * @f$ L_n^\alpha(x) @f$, is defined by</span></div><div class="line"><a name="l00225"></a><span class="lineno"> 225</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00226"></a><span class="lineno"> 226</span> <span class="comment"> * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}</span></div><div class="line"><a name="l00227"></a><span class="lineno"> 227</span> <span class="comment"> * {}_1F_1(-n; \alpha + 1; x)</span></div><div class="line"><a name="l00228"></a><span class="lineno"> 228</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00229"></a><span class="lineno"> 229</span> <span class="comment"> * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and</span></div><div class="line"><a name="l00230"></a><span class="lineno"> 230</span> <span class="comment"> * @f$ {}_1F_1(a; c; x) @f$ is the confluent hypergeometric function.</span></div><div class="line"><a name="l00231"></a><span class="lineno"> 231</span> <span class="comment"> *</span></div><div class="line"><a name="l00232"></a><span class="lineno"> 232</span> <span class="comment"> * The associated Laguerre polynomial is defined for integral</span></div><div class="line"><a name="l00233"></a><span class="lineno"> 233</span> <span class="comment"> * degree @f$ \alpha = m @f$ by:</span></div><div class="line"><a name="l00234"></a><span class="lineno"> 234</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00235"></a><span class="lineno"> 235</span> <span class="comment"> * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)</span></div><div class="line"><a name="l00236"></a><span class="lineno"> 236</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00237"></a><span class="lineno"> 237</span> <span class="comment"> * where the Laguerre polynomial is defined by:</span></div><div class="line"><a name="l00238"></a><span class="lineno"> 238</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00239"></a><span class="lineno"> 239</span> <span class="comment"> * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})</span></div><div class="line"><a name="l00240"></a><span class="lineno"> 240</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00241"></a><span class="lineno"> 241</span> <span class="comment"> * and @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00242"></a><span class="lineno"> 242</span> <span class="comment"> * @see laguerre for details of the Laguerre function of degree @c n</span></div><div class="line"><a name="l00243"></a><span class="lineno"> 243</span> <span class="comment"> *</span></div><div class="line"><a name="l00244"></a><span class="lineno"> 244</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l00245"></a><span class="lineno"> 245</span> <span class="comment"> * @param __n The order of the Laguerre function, <tt>__n >= 0</tt>.</span></div><div class="line"><a name="l00246"></a><span class="lineno"> 246</span> <span class="comment"> * @param __m The degree of the Laguerre function, <tt>__m >= 0</tt>.</span></div><div class="line"><a name="l00247"></a><span class="lineno"> 247</span> <span class="comment"> * @param __x The argument of the Laguerre function, <tt>__x >= 0</tt>.</span></div><div class="line"><a name="l00248"></a><span class="lineno"> 248</span> <span class="comment"> * @throw std::domain_error if <tt>__x < 0</tt>.</span></div><div class="line"><a name="l00249"></a><span class="lineno"> 249</span> <span class="comment"> */</span></div><div class="line"><a name="l00250"></a><span class="lineno"> 250</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00251"></a><span class="lineno"> 251</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l00252"></a><span class="lineno"><a class="line" href="a01497.html#ga377bb7e038c464a27dfe0573fd2d7b33"> 252</a></span>  <a class="code" href="a01497.html#ga377bb7e038c464a27dfe0573fd2d7b33">assoc_laguerre</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __m, _Tp __x)</div><div class="line"><a name="l00253"></a><span class="lineno"> 253</span>  {</div><div class="line"><a name="l00254"></a><span class="lineno"> 254</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l00255"></a><span class="lineno"> 255</span>  <span class="keywordflow">return</span> __detail::__assoc_laguerre<__type>(__n, __m, __x);</div><div class="line"><a name="l00256"></a><span class="lineno"> 256</span>  }</div><div class="line"><a name="l00257"></a><span class="lineno"> 257</span> </div><div class="line"><a name="l00258"></a><span class="lineno"> 258</span>  <span class="comment">// Associated Legendre functions</span></div><div class="line"><a name="l00259"></a><span class="lineno"> 259</span> <span class="comment"></span></div><div class="line"><a name="l00260"></a><span class="lineno"> 260</span> <span class="comment"> /**</span></div><div class="line"><a name="l00261"></a><span class="lineno"> 261</span> <span class="comment"> * Return the associated Legendre function of degree @c l and order @c m</span></div><div class="line"><a name="l00262"></a><span class="lineno"> 262</span> <span class="comment"> * for @c float argument.</span></div><div class="line"><a name="l00263"></a><span class="lineno"> 263</span> <span class="comment"> *</span></div><div class="line"><a name="l00264"></a><span class="lineno"> 264</span> <span class="comment"> * @see assoc_legendre for more details.</span></div><div class="line"><a name="l00265"></a><span class="lineno"> 265</span> <span class="comment"> */</span></div><div class="line"><a name="l00266"></a><span class="lineno"> 266</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00267"></a><span class="lineno"><a class="line" href="a01497.html#ga3ced07ddd24bf4af56e2712d148e7f57"> 267</a></span>  <a class="code" href="a01497.html#ga3ced07ddd24bf4af56e2712d148e7f57">assoc_legendref</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __l, <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __m, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l00268"></a><span class="lineno"> 268</span>  { <span class="keywordflow">return</span> __detail::__assoc_legendre_p<float>(__l, __m, __x); }</div><div class="line"><a name="l00269"></a><span class="lineno"> 269</span> <span class="comment"></span></div><div class="line"><a name="l00270"></a><span class="lineno"> 270</span> <span class="comment"> /**</span></div><div class="line"><a name="l00271"></a><span class="lineno"> 271</span> <span class="comment"> * Return the associated Legendre function of degree @c l and order @c m.</span></div><div class="line"><a name="l00272"></a><span class="lineno"> 272</span> <span class="comment"> *</span></div><div class="line"><a name="l00273"></a><span class="lineno"> 273</span> <span class="comment"> * @see assoc_legendre for more details.</span></div><div class="line"><a name="l00274"></a><span class="lineno"> 274</span> <span class="comment"> */</span></div><div class="line"><a name="l00275"></a><span class="lineno"> 275</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00276"></a><span class="lineno"><a class="line" href="a01497.html#ga55977b425a539146f060dec1c8003344"> 276</a></span>  <a class="code" href="a01497.html#ga55977b425a539146f060dec1c8003344">assoc_legendrel</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __l, <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __m, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l00277"></a><span class="lineno"> 277</span>  { <span class="keywordflow">return</span> __detail::__assoc_legendre_p<long double>(__l, __m, __x); }</div><div class="line"><a name="l00278"></a><span class="lineno"> 278</span> </div><div class="line"><a name="l00279"></a><span class="lineno"> 279</span> <span class="comment"></span></div><div class="line"><a name="l00280"></a><span class="lineno"> 280</span> <span class="comment"> /**</span></div><div class="line"><a name="l00281"></a><span class="lineno"> 281</span> <span class="comment"> * Return the associated Legendre function of degree @c l and order @c m.</span></div><div class="line"><a name="l00282"></a><span class="lineno"> 282</span> <span class="comment"> *</span></div><div class="line"><a name="l00283"></a><span class="lineno"> 283</span> <span class="comment"> * The associated Legendre function is derived from the Legendre function</span></div><div class="line"><a name="l00284"></a><span class="lineno"> 284</span> <span class="comment"> * @f$ P_l(x) @f$ by the Rodrigues formula:</span></div><div class="line"><a name="l00285"></a><span class="lineno"> 285</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00286"></a><span class="lineno"> 286</span> <span class="comment"> * P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)</span></div><div class="line"><a name="l00287"></a><span class="lineno"> 287</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00288"></a><span class="lineno"> 288</span> <span class="comment"> * @see legendre for details of the Legendre function of degree @c l</span></div><div class="line"><a name="l00289"></a><span class="lineno"> 289</span> <span class="comment"> *</span></div><div class="line"><a name="l00290"></a><span class="lineno"> 290</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l00291"></a><span class="lineno"> 291</span> <span class="comment"> * @param __l The degree <tt>__l >= 0</tt>.</span></div><div class="line"><a name="l00292"></a><span class="lineno"> 292</span> <span class="comment"> * @param __m The order <tt>__m <= l</tt>.</span></div><div class="line"><a name="l00293"></a><span class="lineno"> 293</span> <span class="comment"> * @param __x The argument, <tt>abs(__x) <= 1</tt>.</span></div><div class="line"><a name="l00294"></a><span class="lineno"> 294</span> <span class="comment"> * @throw std::domain_error if <tt>abs(__x) > 1</tt>.</span></div><div class="line"><a name="l00295"></a><span class="lineno"> 295</span> <span class="comment"> */</span></div><div class="line"><a name="l00296"></a><span class="lineno"> 296</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00297"></a><span class="lineno"> 297</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l00298"></a><span class="lineno"><a class="line" href="a01497.html#ga355349f79119c1fd1e2a9351cec57f0f"> 298</a></span>  <a class="code" href="a01497.html#ga355349f79119c1fd1e2a9351cec57f0f">assoc_legendre</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __l, <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __m, _Tp __x)</div><div class="line"><a name="l00299"></a><span class="lineno"> 299</span>  {</div><div class="line"><a name="l00300"></a><span class="lineno"> 300</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l00301"></a><span class="lineno"> 301</span>  <span class="keywordflow">return</span> __detail::__assoc_legendre_p<__type>(__l, __m, __x);</div><div class="line"><a name="l00302"></a><span class="lineno"> 302</span>  }</div><div class="line"><a name="l00303"></a><span class="lineno"> 303</span> </div><div class="line"><a name="l00304"></a><span class="lineno"> 304</span>  <span class="comment">// Beta functions</span></div><div class="line"><a name="l00305"></a><span class="lineno"> 305</span> <span class="comment"></span></div><div class="line"><a name="l00306"></a><span class="lineno"> 306</span> <span class="comment"> /**</span></div><div class="line"><a name="l00307"></a><span class="lineno"> 307</span> <span class="comment"> * Return the beta function, @f$ B(a,b) @f$, for @c float parameters @c a, @c b.</span></div><div class="line"><a name="l00308"></a><span class="lineno"> 308</span> <span class="comment"> *</span></div><div class="line"><a name="l00309"></a><span class="lineno"> 309</span> <span class="comment"> * @see beta for more details.</span></div><div class="line"><a name="l00310"></a><span class="lineno"> 310</span> <span class="comment"> */</span></div><div class="line"><a name="l00311"></a><span class="lineno"> 311</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00312"></a><span class="lineno"><a class="line" href="a01497.html#ga12dc61ee4c09172151cf092ed387e203"> 312</a></span>  <a class="code" href="a01497.html#ga12dc61ee4c09172151cf092ed387e203">betaf</a>(<span class="keywordtype">float</span> __a, <span class="keywordtype">float</span> __b)</div><div class="line"><a name="l00313"></a><span class="lineno"> 313</span>  { <span class="keywordflow">return</span> __detail::__beta<float>(__a, __b); }</div><div class="line"><a name="l00314"></a><span class="lineno"> 314</span> <span class="comment"></span></div><div class="line"><a name="l00315"></a><span class="lineno"> 315</span> <span class="comment"> /**</span></div><div class="line"><a name="l00316"></a><span class="lineno"> 316</span> <span class="comment"> * Return the beta function, @f$B(a,b)@f$, for long double</span></div><div class="line"><a name="l00317"></a><span class="lineno"> 317</span> <span class="comment"> * parameters @c a, @c b.</span></div><div class="line"><a name="l00318"></a><span class="lineno"> 318</span> <span class="comment"> *</span></div><div class="line"><a name="l00319"></a><span class="lineno"> 319</span> <span class="comment"> * @see beta for more details.</span></div><div class="line"><a name="l00320"></a><span class="lineno"> 320</span> <span class="comment"> */</span></div><div class="line"><a name="l00321"></a><span class="lineno"> 321</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00322"></a><span class="lineno"><a class="line" href="a01497.html#ga8caca1cef099f41a88111209c36ce06c"> 322</a></span>  <a class="code" href="a01497.html#ga8caca1cef099f41a88111209c36ce06c">betal</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __a, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __b)</div><div class="line"><a name="l00323"></a><span class="lineno"> 323</span>  { <span class="keywordflow">return</span> __detail::__beta<long double>(__a, __b); }</div><div class="line"><a name="l00324"></a><span class="lineno"> 324</span> <span class="comment"></span></div><div class="line"><a name="l00325"></a><span class="lineno"> 325</span> <span class="comment"> /**</span></div><div class="line"><a name="l00326"></a><span class="lineno"> 326</span> <span class="comment"> * Return the beta function, @f$B(a,b)@f$, for real parameters @c a, @c b.</span></div><div class="line"><a name="l00327"></a><span class="lineno"> 327</span> <span class="comment"> *</span></div><div class="line"><a name="l00328"></a><span class="lineno"> 328</span> <span class="comment"> * The beta function is defined by</span></div><div class="line"><a name="l00329"></a><span class="lineno"> 329</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00330"></a><span class="lineno"> 330</span> <span class="comment"> * B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt</span></div><div class="line"><a name="l00331"></a><span class="lineno"> 331</span> <span class="comment"> * = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}</span></div><div class="line"><a name="l00332"></a><span class="lineno"> 332</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00333"></a><span class="lineno"> 333</span> <span class="comment"> * where @f$ a > 0 @f$ and @f$ b > 0 @f$</span></div><div class="line"><a name="l00334"></a><span class="lineno"> 334</span> <span class="comment"> *</span></div><div class="line"><a name="l00335"></a><span class="lineno"> 335</span> <span class="comment"> * @tparam _Tpa The floating-point type of the parameter @c __a.</span></div><div class="line"><a name="l00336"></a><span class="lineno"> 336</span> <span class="comment"> * @tparam _Tpb The floating-point type of the parameter @c __b.</span></div><div class="line"><a name="l00337"></a><span class="lineno"> 337</span> <span class="comment"> * @param __a The first argument of the beta function, <tt> __a > 0 </tt>.</span></div><div class="line"><a name="l00338"></a><span class="lineno"> 338</span> <span class="comment"> * @param __b The second argument of the beta function, <tt> __b > 0 </tt>.</span></div><div class="line"><a name="l00339"></a><span class="lineno"> 339</span> <span class="comment"> * @throw std::domain_error if <tt> __a < 0 </tt> or <tt> __b < 0 </tt>.</span></div><div class="line"><a name="l00340"></a><span class="lineno"> 340</span> <span class="comment"> */</span></div><div class="line"><a name="l00341"></a><span class="lineno"> 341</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tpa, <span class="keyword">typename</span> _Tpb></div><div class="line"><a name="l00342"></a><span class="lineno"> 342</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type</div><div class="line"><a name="l00343"></a><span class="lineno"><a class="line" href="a01497.html#ga6a7220c87c942db48b18b527d92bbd2d"> 343</a></span>  <a class="code" href="a01497.html#ga6a7220c87c942db48b18b527d92bbd2d">beta</a>(_Tpa __a, _Tpb __b)</div><div class="line"><a name="l00344"></a><span class="lineno"> 344</span>  {</div><div class="line"><a name="l00345"></a><span class="lineno"> 345</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type __type;</div><div class="line"><a name="l00346"></a><span class="lineno"> 346</span>  <span class="keywordflow">return</span> __detail::__beta<__type>(__a, __b);</div><div class="line"><a name="l00347"></a><span class="lineno"> 347</span>  }</div><div class="line"><a name="l00348"></a><span class="lineno"> 348</span> </div><div class="line"><a name="l00349"></a><span class="lineno"> 349</span>  <span class="comment">// Complete elliptic integrals of the first kind</span></div><div class="line"><a name="l00350"></a><span class="lineno"> 350</span> <span class="comment"></span></div><div class="line"><a name="l00351"></a><span class="lineno"> 351</span> <span class="comment"> /**</span></div><div class="line"><a name="l00352"></a><span class="lineno"> 352</span> <span class="comment"> * Return the complete elliptic integral of the first kind @f$ E(k) @f$</span></div><div class="line"><a name="l00353"></a><span class="lineno"> 353</span> <span class="comment"> * for @c float modulus @c k.</span></div><div class="line"><a name="l00354"></a><span class="lineno"> 354</span> <span class="comment"> *</span></div><div class="line"><a name="l00355"></a><span class="lineno"> 355</span> <span class="comment"> * @see comp_ellint_1 for details.</span></div><div class="line"><a name="l00356"></a><span class="lineno"> 356</span> <span class="comment"> */</span></div><div class="line"><a name="l00357"></a><span class="lineno"> 357</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00358"></a><span class="lineno"><a class="line" href="a01497.html#ga7fb5be999a8125cf7e55e630eb8444a1"> 358</a></span>  <a class="code" href="a01497.html#ga7fb5be999a8125cf7e55e630eb8444a1">comp_ellint_1f</a>(<span class="keywordtype">float</span> __k)</div><div class="line"><a name="l00359"></a><span class="lineno"> 359</span>  { <span class="keywordflow">return</span> __detail::__comp_ellint_1<float>(__k); }</div><div class="line"><a name="l00360"></a><span class="lineno"> 360</span> <span class="comment"></span></div><div class="line"><a name="l00361"></a><span class="lineno"> 361</span> <span class="comment"> /**</span></div><div class="line"><a name="l00362"></a><span class="lineno"> 362</span> <span class="comment"> * Return the complete elliptic integral of the first kind @f$ E(k) @f$</span></div><div class="line"><a name="l00363"></a><span class="lineno"> 363</span> <span class="comment"> * for long double modulus @c k.</span></div><div class="line"><a name="l00364"></a><span class="lineno"> 364</span> <span class="comment"> *</span></div><div class="line"><a name="l00365"></a><span class="lineno"> 365</span> <span class="comment"> * @see comp_ellint_1 for details.</span></div><div class="line"><a name="l00366"></a><span class="lineno"> 366</span> <span class="comment"> */</span></div><div class="line"><a name="l00367"></a><span class="lineno"> 367</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00368"></a><span class="lineno"><a class="line" href="a01497.html#ga7247d3dd77c1ff5df3c059fed862dc48"> 368</a></span>  <a class="code" href="a01497.html#ga7247d3dd77c1ff5df3c059fed862dc48">comp_ellint_1l</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __k)</div><div class="line"><a name="l00369"></a><span class="lineno"> 369</span>  { <span class="keywordflow">return</span> __detail::__comp_ellint_1<long double>(__k); }</div><div class="line"><a name="l00370"></a><span class="lineno"> 370</span> <span class="comment"></span></div><div class="line"><a name="l00371"></a><span class="lineno"> 371</span> <span class="comment"> /**</span></div><div class="line"><a name="l00372"></a><span class="lineno"> 372</span> <span class="comment"> * Return the complete elliptic integral of the first kind</span></div><div class="line"><a name="l00373"></a><span class="lineno"> 373</span> <span class="comment"> * @f$ K(k) @f$ for real modulus @c k.</span></div><div class="line"><a name="l00374"></a><span class="lineno"> 374</span> <span class="comment"> *</span></div><div class="line"><a name="l00375"></a><span class="lineno"> 375</span> <span class="comment"> * The complete elliptic integral of the first kind is defined as</span></div><div class="line"><a name="l00376"></a><span class="lineno"> 376</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00377"></a><span class="lineno"> 377</span> <span class="comment"> * K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta}</span></div><div class="line"><a name="l00378"></a><span class="lineno"> 378</span> <span class="comment"> * {\sqrt{1 - k^2 sin^2\theta}}</span></div><div class="line"><a name="l00379"></a><span class="lineno"> 379</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00380"></a><span class="lineno"> 380</span> <span class="comment"> * where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the</span></div><div class="line"><a name="l00381"></a><span class="lineno"> 381</span> <span class="comment"> * first kind and the modulus @f$ |k| <= 1 @f$.</span></div><div class="line"><a name="l00382"></a><span class="lineno"> 382</span> <span class="comment"> * @see ellint_1 for details of the incomplete elliptic function</span></div><div class="line"><a name="l00383"></a><span class="lineno"> 383</span> <span class="comment"> * of the first kind.</span></div><div class="line"><a name="l00384"></a><span class="lineno"> 384</span> <span class="comment"> *</span></div><div class="line"><a name="l00385"></a><span class="lineno"> 385</span> <span class="comment"> * @tparam _Tp The floating-point type of the modulus @c __k.</span></div><div class="line"><a name="l00386"></a><span class="lineno"> 386</span> <span class="comment"> * @param __k The modulus, <tt> abs(__k) <= 1 </tt></span></div><div class="line"><a name="l00387"></a><span class="lineno"> 387</span> <span class="comment"> * @throw std::domain_error if <tt> abs(__k) > 1 </tt>.</span></div><div class="line"><a name="l00388"></a><span class="lineno"> 388</span> <span class="comment"> */</span></div><div class="line"><a name="l00389"></a><span class="lineno"> 389</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00390"></a><span class="lineno"> 390</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l00391"></a><span class="lineno"><a class="line" href="a01497.html#gac559500c604c43ea943d593c9ad9d289"> 391</a></span>  <a class="code" href="a01497.html#gac559500c604c43ea943d593c9ad9d289">comp_ellint_1</a>(_Tp __k)</div><div class="line"><a name="l00392"></a><span class="lineno"> 392</span>  {</div><div class="line"><a name="l00393"></a><span class="lineno"> 393</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l00394"></a><span class="lineno"> 394</span>  <span class="keywordflow">return</span> __detail::__comp_ellint_1<__type>(__k);</div><div class="line"><a name="l00395"></a><span class="lineno"> 395</span>  }</div><div class="line"><a name="l00396"></a><span class="lineno"> 396</span> </div><div class="line"><a name="l00397"></a><span class="lineno"> 397</span>  <span class="comment">// Complete elliptic integrals of the second kind</span></div><div class="line"><a name="l00398"></a><span class="lineno"> 398</span> <span class="comment"></span></div><div class="line"><a name="l00399"></a><span class="lineno"> 399</span> <span class="comment"> /**</span></div><div class="line"><a name="l00400"></a><span class="lineno"> 400</span> <span class="comment"> * Return the complete elliptic integral of the second kind @f$ E(k) @f$</span></div><div class="line"><a name="l00401"></a><span class="lineno"> 401</span> <span class="comment"> * for @c float modulus @c k.</span></div><div class="line"><a name="l00402"></a><span class="lineno"> 402</span> <span class="comment"> *</span></div><div class="line"><a name="l00403"></a><span class="lineno"> 403</span> <span class="comment"> * @see comp_ellint_2 for details.</span></div><div class="line"><a name="l00404"></a><span class="lineno"> 404</span> <span class="comment"> */</span></div><div class="line"><a name="l00405"></a><span class="lineno"> 405</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00406"></a><span class="lineno"><a class="line" href="a01497.html#ga21700f2f125c42b1f1da1f9c7eea1135"> 406</a></span>  <a class="code" href="a01497.html#ga21700f2f125c42b1f1da1f9c7eea1135">comp_ellint_2f</a>(<span class="keywordtype">float</span> __k)</div><div class="line"><a name="l00407"></a><span class="lineno"> 407</span>  { <span class="keywordflow">return</span> __detail::__comp_ellint_2<float>(__k); }</div><div class="line"><a name="l00408"></a><span class="lineno"> 408</span> <span class="comment"></span></div><div class="line"><a name="l00409"></a><span class="lineno"> 409</span> <span class="comment"> /**</span></div><div class="line"><a name="l00410"></a><span class="lineno"> 410</span> <span class="comment"> * Return the complete elliptic integral of the second kind @f$ E(k) @f$</span></div><div class="line"><a name="l00411"></a><span class="lineno"> 411</span> <span class="comment"> * for long double modulus @c k.</span></div><div class="line"><a name="l00412"></a><span class="lineno"> 412</span> <span class="comment"> *</span></div><div class="line"><a name="l00413"></a><span class="lineno"> 413</span> <span class="comment"> * @see comp_ellint_2 for details.</span></div><div class="line"><a name="l00414"></a><span class="lineno"> 414</span> <span class="comment"> */</span></div><div class="line"><a name="l00415"></a><span class="lineno"> 415</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00416"></a><span class="lineno"><a class="line" href="a01497.html#ga47b647ec386c8d4b18a030c97842df18"> 416</a></span>  <a class="code" href="a01497.html#ga47b647ec386c8d4b18a030c97842df18">comp_ellint_2l</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __k)</div><div class="line"><a name="l00417"></a><span class="lineno"> 417</span>  { <span class="keywordflow">return</span> __detail::__comp_ellint_2<long double>(__k); }</div><div class="line"><a name="l00418"></a><span class="lineno"> 418</span> <span class="comment"></span></div><div class="line"><a name="l00419"></a><span class="lineno"> 419</span> <span class="comment"> /**</span></div><div class="line"><a name="l00420"></a><span class="lineno"> 420</span> <span class="comment"> * Return the complete elliptic integral of the second kind @f$ E(k) @f$</span></div><div class="line"><a name="l00421"></a><span class="lineno"> 421</span> <span class="comment"> * for real modulus @c k.</span></div><div class="line"><a name="l00422"></a><span class="lineno"> 422</span> <span class="comment"> *</span></div><div class="line"><a name="l00423"></a><span class="lineno"> 423</span> <span class="comment"> * The complete elliptic integral of the second kind is defined as</span></div><div class="line"><a name="l00424"></a><span class="lineno"> 424</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00425"></a><span class="lineno"> 425</span> <span class="comment"> * E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}</span></div><div class="line"><a name="l00426"></a><span class="lineno"> 426</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00427"></a><span class="lineno"> 427</span> <span class="comment"> * where @f$ E(k,\phi) @f$ is the incomplete elliptic integral of the</span></div><div class="line"><a name="l00428"></a><span class="lineno"> 428</span> <span class="comment"> * second kind and the modulus @f$ |k| <= 1 @f$.</span></div><div class="line"><a name="l00429"></a><span class="lineno"> 429</span> <span class="comment"> * @see ellint_2 for details of the incomplete elliptic function</span></div><div class="line"><a name="l00430"></a><span class="lineno"> 430</span> <span class="comment"> * of the second kind.</span></div><div class="line"><a name="l00431"></a><span class="lineno"> 431</span> <span class="comment"> *</span></div><div class="line"><a name="l00432"></a><span class="lineno"> 432</span> <span class="comment"> * @tparam _Tp The floating-point type of the modulus @c __k.</span></div><div class="line"><a name="l00433"></a><span class="lineno"> 433</span> <span class="comment"> * @param __k The modulus, @c abs(__k) <= 1</span></div><div class="line"><a name="l00434"></a><span class="lineno"> 434</span> <span class="comment"> * @throw std::domain_error if @c abs(__k) > 1.</span></div><div class="line"><a name="l00435"></a><span class="lineno"> 435</span> <span class="comment"> */</span></div><div class="line"><a name="l00436"></a><span class="lineno"> 436</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00437"></a><span class="lineno"> 437</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l00438"></a><span class="lineno"><a class="line" href="a01497.html#ga22fcc678829f0daf2de257896378e7e0"> 438</a></span>  <a class="code" href="a01497.html#ga22fcc678829f0daf2de257896378e7e0">comp_ellint_2</a>(_Tp __k)</div><div class="line"><a name="l00439"></a><span class="lineno"> 439</span>  {</div><div class="line"><a name="l00440"></a><span class="lineno"> 440</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l00441"></a><span class="lineno"> 441</span>  <span class="keywordflow">return</span> __detail::__comp_ellint_2<__type>(__k);</div><div class="line"><a name="l00442"></a><span class="lineno"> 442</span>  }</div><div class="line"><a name="l00443"></a><span class="lineno"> 443</span> </div><div class="line"><a name="l00444"></a><span class="lineno"> 444</span>  <span class="comment">// Complete elliptic integrals of the third kind</span></div><div class="line"><a name="l00445"></a><span class="lineno"> 445</span> <span class="comment"></span></div><div class="line"><a name="l00446"></a><span class="lineno"> 446</span> <span class="comment"> /**</span></div><div class="line"><a name="l00447"></a><span class="lineno"> 447</span> <span class="comment"> * @brief Return the complete elliptic integral of the third kind</span></div><div class="line"><a name="l00448"></a><span class="lineno"> 448</span> <span class="comment"> * @f$ \Pi(k,\nu) @f$ for @c float modulus @c k.</span></div><div class="line"><a name="l00449"></a><span class="lineno"> 449</span> <span class="comment"> *</span></div><div class="line"><a name="l00450"></a><span class="lineno"> 450</span> <span class="comment"> * @see comp_ellint_3 for details.</span></div><div class="line"><a name="l00451"></a><span class="lineno"> 451</span> <span class="comment"> */</span></div><div class="line"><a name="l00452"></a><span class="lineno"> 452</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00453"></a><span class="lineno"><a class="line" href="a01497.html#ga76834d3112f777703330892303267a39"> 453</a></span>  <a class="code" href="a01497.html#ga76834d3112f777703330892303267a39">comp_ellint_3f</a>(<span class="keywordtype">float</span> __k, <span class="keywordtype">float</span> __nu)</div><div class="line"><a name="l00454"></a><span class="lineno"> 454</span>  { <span class="keywordflow">return</span> __detail::__comp_ellint_3<float>(__k, __nu); }</div><div class="line"><a name="l00455"></a><span class="lineno"> 455</span> <span class="comment"></span></div><div class="line"><a name="l00456"></a><span class="lineno"> 456</span> <span class="comment"> /**</span></div><div class="line"><a name="l00457"></a><span class="lineno"> 457</span> <span class="comment"> * @brief Return the complete elliptic integral of the third kind</span></div><div class="line"><a name="l00458"></a><span class="lineno"> 458</span> <span class="comment"> * @f$ \Pi(k,\nu) @f$ for <tt>long double</tt> modulus @c k.</span></div><div class="line"><a name="l00459"></a><span class="lineno"> 459</span> <span class="comment"> *</span></div><div class="line"><a name="l00460"></a><span class="lineno"> 460</span> <span class="comment"> * @see comp_ellint_3 for details.</span></div><div class="line"><a name="l00461"></a><span class="lineno"> 461</span> <span class="comment"> */</span></div><div class="line"><a name="l00462"></a><span class="lineno"> 462</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00463"></a><span class="lineno"><a class="line" href="a01497.html#ga1ca081fee102cd0d4d6b091285e495e5"> 463</a></span>  <a class="code" href="a01497.html#ga1ca081fee102cd0d4d6b091285e495e5">comp_ellint_3l</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __k, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __nu)</div><div class="line"><a name="l00464"></a><span class="lineno"> 464</span>  { <span class="keywordflow">return</span> __detail::__comp_ellint_3<long double>(__k, __nu); }</div><div class="line"><a name="l00465"></a><span class="lineno"> 465</span> <span class="comment"></span></div><div class="line"><a name="l00466"></a><span class="lineno"> 466</span> <span class="comment"> /**</span></div><div class="line"><a name="l00467"></a><span class="lineno"> 467</span> <span class="comment"> * Return the complete elliptic integral of the third kind</span></div><div class="line"><a name="l00468"></a><span class="lineno"> 468</span> <span class="comment"> * @f$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) @f$ for real modulus @c k.</span></div><div class="line"><a name="l00469"></a><span class="lineno"> 469</span> <span class="comment"> *</span></div><div class="line"><a name="l00470"></a><span class="lineno"> 470</span> <span class="comment"> * The complete elliptic integral of the third kind is defined as</span></div><div class="line"><a name="l00471"></a><span class="lineno"> 471</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00472"></a><span class="lineno"> 472</span> <span class="comment"> * \Pi(k,\nu) = \Pi(k,\nu,\pi/2) = \int_0^{\pi/2}</span></div><div class="line"><a name="l00473"></a><span class="lineno"> 473</span> <span class="comment"> * \frac{d\theta}</span></div><div class="line"><a name="l00474"></a><span class="lineno"> 474</span> <span class="comment"> * {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}}</span></div><div class="line"><a name="l00475"></a><span class="lineno"> 475</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00476"></a><span class="lineno"> 476</span> <span class="comment"> * where @f$ \Pi(k,\nu,\phi) @f$ is the incomplete elliptic integral of the</span></div><div class="line"><a name="l00477"></a><span class="lineno"> 477</span> <span class="comment"> * second kind and the modulus @f$ |k| <= 1 @f$.</span></div><div class="line"><a name="l00478"></a><span class="lineno"> 478</span> <span class="comment"> * @see ellint_3 for details of the incomplete elliptic function</span></div><div class="line"><a name="l00479"></a><span class="lineno"> 479</span> <span class="comment"> * of the third kind.</span></div><div class="line"><a name="l00480"></a><span class="lineno"> 480</span> <span class="comment"> *</span></div><div class="line"><a name="l00481"></a><span class="lineno"> 481</span> <span class="comment"> * @tparam _Tp The floating-point type of the modulus @c __k.</span></div><div class="line"><a name="l00482"></a><span class="lineno"> 482</span> <span class="comment"> * @tparam _Tpn The floating-point type of the argument @c __nu.</span></div><div class="line"><a name="l00483"></a><span class="lineno"> 483</span> <span class="comment"> * @param __k The modulus, @c abs(__k) <= 1</span></div><div class="line"><a name="l00484"></a><span class="lineno"> 484</span> <span class="comment"> * @param __nu The argument</span></div><div class="line"><a name="l00485"></a><span class="lineno"> 485</span> <span class="comment"> * @throw std::domain_error if @c abs(__k) > 1.</span></div><div class="line"><a name="l00486"></a><span class="lineno"> 486</span> <span class="comment"> */</span></div><div class="line"><a name="l00487"></a><span class="lineno"> 487</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp, <span class="keyword">typename</span> _Tpn></div><div class="line"><a name="l00488"></a><span class="lineno"> 488</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tp, _Tpn>::__type</div><div class="line"><a name="l00489"></a><span class="lineno"><a class="line" href="a01497.html#gad833404645e24b7f0598a640ff92d623"> 489</a></span>  <a class="code" href="a01497.html#gad833404645e24b7f0598a640ff92d623">comp_ellint_3</a>(_Tp __k, _Tpn __nu)</div><div class="line"><a name="l00490"></a><span class="lineno"> 490</span>  {</div><div class="line"><a name="l00491"></a><span class="lineno"> 491</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tp, _Tpn>::__type __type;</div><div class="line"><a name="l00492"></a><span class="lineno"> 492</span>  <span class="keywordflow">return</span> __detail::__comp_ellint_3<__type>(__k, __nu);</div><div class="line"><a name="l00493"></a><span class="lineno"> 493</span>  }</div><div class="line"><a name="l00494"></a><span class="lineno"> 494</span> </div><div class="line"><a name="l00495"></a><span class="lineno"> 495</span>  <span class="comment">// Regular modified cylindrical Bessel functions</span></div><div class="line"><a name="l00496"></a><span class="lineno"> 496</span> <span class="comment"></span></div><div class="line"><a name="l00497"></a><span class="lineno"> 497</span> <span class="comment"> /**</span></div><div class="line"><a name="l00498"></a><span class="lineno"> 498</span> <span class="comment"> * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$</span></div><div class="line"><a name="l00499"></a><span class="lineno"> 499</span> <span class="comment"> * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00500"></a><span class="lineno"> 500</span> <span class="comment"> *</span></div><div class="line"><a name="l00501"></a><span class="lineno"> 501</span> <span class="comment"> * @see cyl_bessel_i for setails.</span></div><div class="line"><a name="l00502"></a><span class="lineno"> 502</span> <span class="comment"> */</span></div><div class="line"><a name="l00503"></a><span class="lineno"> 503</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00504"></a><span class="lineno"><a class="line" href="a01497.html#gaaf738427d4da0bda66bc2274dfb853a7"> 504</a></span>  <a class="code" href="a01497.html#gaaf738427d4da0bda66bc2274dfb853a7">cyl_bessel_if</a>(<span class="keywordtype">float</span> __nu, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l00505"></a><span class="lineno"> 505</span>  { <span class="keywordflow">return</span> __detail::__cyl_bessel_i<float>(__nu, __x); }</div><div class="line"><a name="l00506"></a><span class="lineno"> 506</span> <span class="comment"></span></div><div class="line"><a name="l00507"></a><span class="lineno"> 507</span> <span class="comment"> /**</span></div><div class="line"><a name="l00508"></a><span class="lineno"> 508</span> <span class="comment"> * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$</span></div><div class="line"><a name="l00509"></a><span class="lineno"> 509</span> <span class="comment"> * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00510"></a><span class="lineno"> 510</span> <span class="comment"> *</span></div><div class="line"><a name="l00511"></a><span class="lineno"> 511</span> <span class="comment"> * @see cyl_bessel_i for setails.</span></div><div class="line"><a name="l00512"></a><span class="lineno"> 512</span> <span class="comment"> */</span></div><div class="line"><a name="l00513"></a><span class="lineno"> 513</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00514"></a><span class="lineno"><a class="line" href="a01497.html#gab7962629216d03efb8ecaa3f70c6878f"> 514</a></span>  <a class="code" href="a01497.html#gab7962629216d03efb8ecaa3f70c6878f">cyl_bessel_il</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __nu, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l00515"></a><span class="lineno"> 515</span>  { <span class="keywordflow">return</span> __detail::__cyl_bessel_i<long double>(__nu, __x); }</div><div class="line"><a name="l00516"></a><span class="lineno"> 516</span> <span class="comment"></span></div><div class="line"><a name="l00517"></a><span class="lineno"> 517</span> <span class="comment"> /**</span></div><div class="line"><a name="l00518"></a><span class="lineno"> 518</span> <span class="comment"> * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$</span></div><div class="line"><a name="l00519"></a><span class="lineno"> 519</span> <span class="comment"> * for real order @f$ \nu @f$ and argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00520"></a><span class="lineno"> 520</span> <span class="comment"> *</span></div><div class="line"><a name="l00521"></a><span class="lineno"> 521</span> <span class="comment"> * The regular modified cylindrical Bessel function is:</span></div><div class="line"><a name="l00522"></a><span class="lineno"> 522</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00523"></a><span class="lineno"> 523</span> <span class="comment"> * I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty}</span></div><div class="line"><a name="l00524"></a><span class="lineno"> 524</span> <span class="comment"> * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}</span></div><div class="line"><a name="l00525"></a><span class="lineno"> 525</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00526"></a><span class="lineno"> 526</span> <span class="comment"> *</span></div><div class="line"><a name="l00527"></a><span class="lineno"> 527</span> <span class="comment"> * @tparam _Tpnu The floating-point type of the order @c __nu.</span></div><div class="line"><a name="l00528"></a><span class="lineno"> 528</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l00529"></a><span class="lineno"> 529</span> <span class="comment"> * @param __nu The order</span></div><div class="line"><a name="l00530"></a><span class="lineno"> 530</span> <span class="comment"> * @param __x The argument, <tt> __x >= 0 </tt></span></div><div class="line"><a name="l00531"></a><span class="lineno"> 531</span> <span class="comment"> * @throw std::domain_error if <tt> __x < 0 </tt>.</span></div><div class="line"><a name="l00532"></a><span class="lineno"> 532</span> <span class="comment"> */</span></div><div class="line"><a name="l00533"></a><span class="lineno"> 533</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tpnu, <span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00534"></a><span class="lineno"> 534</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type</div><div class="line"><a name="l00535"></a><span class="lineno"><a class="line" href="a01497.html#ga1c9b5a5c36f000a4f0a55f7fcc486cb0"> 535</a></span>  <a class="code" href="a01497.html#ga1c9b5a5c36f000a4f0a55f7fcc486cb0">cyl_bessel_i</a>(_Tpnu __nu, _Tp __x)</div><div class="line"><a name="l00536"></a><span class="lineno"> 536</span>  {</div><div class="line"><a name="l00537"></a><span class="lineno"> 537</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type;</div><div class="line"><a name="l00538"></a><span class="lineno"> 538</span>  <span class="keywordflow">return</span> __detail::__cyl_bessel_i<__type>(__nu, __x);</div><div class="line"><a name="l00539"></a><span class="lineno"> 539</span>  }</div><div class="line"><a name="l00540"></a><span class="lineno"> 540</span> </div><div class="line"><a name="l00541"></a><span class="lineno"> 541</span>  <span class="comment">// Cylindrical Bessel functions (of the first kind)</span></div><div class="line"><a name="l00542"></a><span class="lineno"> 542</span> <span class="comment"></span></div><div class="line"><a name="l00543"></a><span class="lineno"> 543</span> <span class="comment"> /**</span></div><div class="line"><a name="l00544"></a><span class="lineno"> 544</span> <span class="comment"> * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$</span></div><div class="line"><a name="l00545"></a><span class="lineno"> 545</span> <span class="comment"> * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00546"></a><span class="lineno"> 546</span> <span class="comment"> *</span></div><div class="line"><a name="l00547"></a><span class="lineno"> 547</span> <span class="comment"> * @see cyl_bessel_j for setails.</span></div><div class="line"><a name="l00548"></a><span class="lineno"> 548</span> <span class="comment"> */</span></div><div class="line"><a name="l00549"></a><span class="lineno"> 549</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00550"></a><span class="lineno"><a class="line" href="a01497.html#ga15731a7bccd6351d28353e3c4c2a2d23"> 550</a></span>  <a class="code" href="a01497.html#ga15731a7bccd6351d28353e3c4c2a2d23">cyl_bessel_jf</a>(<span class="keywordtype">float</span> __nu, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l00551"></a><span class="lineno"> 551</span>  { <span class="keywordflow">return</span> __detail::__cyl_bessel_j<float>(__nu, __x); }</div><div class="line"><a name="l00552"></a><span class="lineno"> 552</span> <span class="comment"></span></div><div class="line"><a name="l00553"></a><span class="lineno"> 553</span> <span class="comment"> /**</span></div><div class="line"><a name="l00554"></a><span class="lineno"> 554</span> <span class="comment"> * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$</span></div><div class="line"><a name="l00555"></a><span class="lineno"> 555</span> <span class="comment"> * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00556"></a><span class="lineno"> 556</span> <span class="comment"> *</span></div><div class="line"><a name="l00557"></a><span class="lineno"> 557</span> <span class="comment"> * @see cyl_bessel_j for setails.</span></div><div class="line"><a name="l00558"></a><span class="lineno"> 558</span> <span class="comment"> */</span></div><div class="line"><a name="l00559"></a><span class="lineno"> 559</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00560"></a><span class="lineno"><a class="line" href="a01497.html#gade8e94a80520a8b628b2d658755b25c0"> 560</a></span>  <a class="code" href="a01497.html#gade8e94a80520a8b628b2d658755b25c0">cyl_bessel_jl</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __nu, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l00561"></a><span class="lineno"> 561</span>  { <span class="keywordflow">return</span> __detail::__cyl_bessel_j<long double>(__nu, __x); }</div><div class="line"><a name="l00562"></a><span class="lineno"> 562</span> <span class="comment"></span></div><div class="line"><a name="l00563"></a><span class="lineno"> 563</span> <span class="comment"> /**</span></div><div class="line"><a name="l00564"></a><span class="lineno"> 564</span> <span class="comment"> * Return the Bessel function @f$ J_{\nu}(x) @f$ of real order @f$ \nu @f$</span></div><div class="line"><a name="l00565"></a><span class="lineno"> 565</span> <span class="comment"> * and argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00566"></a><span class="lineno"> 566</span> <span class="comment"> *</span></div><div class="line"><a name="l00567"></a><span class="lineno"> 567</span> <span class="comment"> * The cylindrical Bessel function is:</span></div><div class="line"><a name="l00568"></a><span class="lineno"> 568</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00569"></a><span class="lineno"> 569</span> <span class="comment"> * J_{\nu}(x) = \sum_{k=0}^{\infty}</span></div><div class="line"><a name="l00570"></a><span class="lineno"> 570</span> <span class="comment"> * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)}</span></div><div class="line"><a name="l00571"></a><span class="lineno"> 571</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00572"></a><span class="lineno"> 572</span> <span class="comment"> *</span></div><div class="line"><a name="l00573"></a><span class="lineno"> 573</span> <span class="comment"> * @tparam _Tpnu The floating-point type of the order @c __nu.</span></div><div class="line"><a name="l00574"></a><span class="lineno"> 574</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l00575"></a><span class="lineno"> 575</span> <span class="comment"> * @param __nu The order</span></div><div class="line"><a name="l00576"></a><span class="lineno"> 576</span> <span class="comment"> * @param __x The argument, <tt> __x >= 0 </tt></span></div><div class="line"><a name="l00577"></a><span class="lineno"> 577</span> <span class="comment"> * @throw std::domain_error if <tt> __x < 0 </tt>.</span></div><div class="line"><a name="l00578"></a><span class="lineno"> 578</span> <span class="comment"> */</span></div><div class="line"><a name="l00579"></a><span class="lineno"> 579</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tpnu, <span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00580"></a><span class="lineno"> 580</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type</div><div class="line"><a name="l00581"></a><span class="lineno"><a class="line" href="a01497.html#ga47e21a13b6d68d0d7f057699bd3b3ce0"> 581</a></span>  <a class="code" href="a01497.html#ga47e21a13b6d68d0d7f057699bd3b3ce0">cyl_bessel_j</a>(_Tpnu __nu, _Tp __x)</div><div class="line"><a name="l00582"></a><span class="lineno"> 582</span>  {</div><div class="line"><a name="l00583"></a><span class="lineno"> 583</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type;</div><div class="line"><a name="l00584"></a><span class="lineno"> 584</span>  <span class="keywordflow">return</span> __detail::__cyl_bessel_j<__type>(__nu, __x);</div><div class="line"><a name="l00585"></a><span class="lineno"> 585</span>  }</div><div class="line"><a name="l00586"></a><span class="lineno"> 586</span> </div><div class="line"><a name="l00587"></a><span class="lineno"> 587</span>  <span class="comment">// Irregular modified cylindrical Bessel functions</span></div><div class="line"><a name="l00588"></a><span class="lineno"> 588</span> <span class="comment"></span></div><div class="line"><a name="l00589"></a><span class="lineno"> 589</span> <span class="comment"> /**</span></div><div class="line"><a name="l00590"></a><span class="lineno"> 590</span> <span class="comment"> * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$</span></div><div class="line"><a name="l00591"></a><span class="lineno"> 591</span> <span class="comment"> * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00592"></a><span class="lineno"> 592</span> <span class="comment"> *</span></div><div class="line"><a name="l00593"></a><span class="lineno"> 593</span> <span class="comment"> * @see cyl_bessel_k for setails.</span></div><div class="line"><a name="l00594"></a><span class="lineno"> 594</span> <span class="comment"> */</span></div><div class="line"><a name="l00595"></a><span class="lineno"> 595</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00596"></a><span class="lineno"><a class="line" href="a01497.html#ga1f50047f9aab0ec8b1a1615fe9fbe32f"> 596</a></span>  <a class="code" href="a01497.html#ga1f50047f9aab0ec8b1a1615fe9fbe32f">cyl_bessel_kf</a>(<span class="keywordtype">float</span> __nu, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l00597"></a><span class="lineno"> 597</span>  { <span class="keywordflow">return</span> __detail::__cyl_bessel_k<float>(__nu, __x); }</div><div class="line"><a name="l00598"></a><span class="lineno"> 598</span> <span class="comment"></span></div><div class="line"><a name="l00599"></a><span class="lineno"> 599</span> <span class="comment"> /**</span></div><div class="line"><a name="l00600"></a><span class="lineno"> 600</span> <span class="comment"> * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$</span></div><div class="line"><a name="l00601"></a><span class="lineno"> 601</span> <span class="comment"> * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00602"></a><span class="lineno"> 602</span> <span class="comment"> *</span></div><div class="line"><a name="l00603"></a><span class="lineno"> 603</span> <span class="comment"> * @see cyl_bessel_k for setails.</span></div><div class="line"><a name="l00604"></a><span class="lineno"> 604</span> <span class="comment"> */</span></div><div class="line"><a name="l00605"></a><span class="lineno"> 605</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00606"></a><span class="lineno"><a class="line" href="a01497.html#gac35194b926270d7857d651e06198c7d3"> 606</a></span>  <a class="code" href="a01497.html#gac35194b926270d7857d651e06198c7d3">cyl_bessel_kl</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __nu, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l00607"></a><span class="lineno"> 607</span>  { <span class="keywordflow">return</span> __detail::__cyl_bessel_k<long double>(__nu, __x); }</div><div class="line"><a name="l00608"></a><span class="lineno"> 608</span> <span class="comment"></span></div><div class="line"><a name="l00609"></a><span class="lineno"> 609</span> <span class="comment"> /**</span></div><div class="line"><a name="l00610"></a><span class="lineno"> 610</span> <span class="comment"> * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$</span></div><div class="line"><a name="l00611"></a><span class="lineno"> 611</span> <span class="comment"> * of real order @f$ \nu @f$ and argument @f$ x @f$.</span></div><div class="line"><a name="l00612"></a><span class="lineno"> 612</span> <span class="comment"> *</span></div><div class="line"><a name="l00613"></a><span class="lineno"> 613</span> <span class="comment"> * The irregular modified Bessel function is defined by:</span></div><div class="line"><a name="l00614"></a><span class="lineno"> 614</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00615"></a><span class="lineno"> 615</span> <span class="comment"> * K_{\nu}(x) = \frac{\pi}{2}</span></div><div class="line"><a name="l00616"></a><span class="lineno"> 616</span> <span class="comment"> * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi}</span></div><div class="line"><a name="l00617"></a><span class="lineno"> 617</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00618"></a><span class="lineno"> 618</span> <span class="comment"> * where for integral @f$ \nu = n @f$ a limit is taken:</span></div><div class="line"><a name="l00619"></a><span class="lineno"> 619</span> <span class="comment"> * @f$ lim_{\nu \to n} @f$.</span></div><div class="line"><a name="l00620"></a><span class="lineno"> 620</span> <span class="comment"> * For negative argument we have simply:</span></div><div class="line"><a name="l00621"></a><span class="lineno"> 621</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00622"></a><span class="lineno"> 622</span> <span class="comment"> * K_{-\nu}(x) = K_{\nu}(x)</span></div><div class="line"><a name="l00623"></a><span class="lineno"> 623</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00624"></a><span class="lineno"> 624</span> <span class="comment"> *</span></div><div class="line"><a name="l00625"></a><span class="lineno"> 625</span> <span class="comment"> * @tparam _Tpnu The floating-point type of the order @c __nu.</span></div><div class="line"><a name="l00626"></a><span class="lineno"> 626</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l00627"></a><span class="lineno"> 627</span> <span class="comment"> * @param __nu The order</span></div><div class="line"><a name="l00628"></a><span class="lineno"> 628</span> <span class="comment"> * @param __x The argument, <tt> __x >= 0 </tt></span></div><div class="line"><a name="l00629"></a><span class="lineno"> 629</span> <span class="comment"> * @throw std::domain_error if <tt> __x < 0 </tt>.</span></div><div class="line"><a name="l00630"></a><span class="lineno"> 630</span> <span class="comment"> */</span></div><div class="line"><a name="l00631"></a><span class="lineno"> 631</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tpnu, <span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00632"></a><span class="lineno"> 632</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type</div><div class="line"><a name="l00633"></a><span class="lineno"><a class="line" href="a01497.html#ga76dcd3884620955680112aca0d327ada"> 633</a></span>  <a class="code" href="a01497.html#ga76dcd3884620955680112aca0d327ada">cyl_bessel_k</a>(_Tpnu __nu, _Tp __x)</div><div class="line"><a name="l00634"></a><span class="lineno"> 634</span>  {</div><div class="line"><a name="l00635"></a><span class="lineno"> 635</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type;</div><div class="line"><a name="l00636"></a><span class="lineno"> 636</span>  <span class="keywordflow">return</span> __detail::__cyl_bessel_k<__type>(__nu, __x);</div><div class="line"><a name="l00637"></a><span class="lineno"> 637</span>  }</div><div class="line"><a name="l00638"></a><span class="lineno"> 638</span> </div><div class="line"><a name="l00639"></a><span class="lineno"> 639</span>  <span class="comment">// Cylindrical Neumann functions</span></div><div class="line"><a name="l00640"></a><span class="lineno"> 640</span> <span class="comment"></span></div><div class="line"><a name="l00641"></a><span class="lineno"> 641</span> <span class="comment"> /**</span></div><div class="line"><a name="l00642"></a><span class="lineno"> 642</span> <span class="comment"> * Return the Neumann function @f$ N_{\nu}(x) @f$</span></div><div class="line"><a name="l00643"></a><span class="lineno"> 643</span> <span class="comment"> * of @c float order @f$ \nu @f$ and argument @f$ x @f$.</span></div><div class="line"><a name="l00644"></a><span class="lineno"> 644</span> <span class="comment"> *</span></div><div class="line"><a name="l00645"></a><span class="lineno"> 645</span> <span class="comment"> * @see cyl_neumann for setails.</span></div><div class="line"><a name="l00646"></a><span class="lineno"> 646</span> <span class="comment"> */</span></div><div class="line"><a name="l00647"></a><span class="lineno"> 647</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00648"></a><span class="lineno"><a class="line" href="a01497.html#ga604c13e8f2bb7cd3c7c91d8b19d6b13a"> 648</a></span>  <a class="code" href="a01497.html#ga604c13e8f2bb7cd3c7c91d8b19d6b13a">cyl_neumannf</a>(<span class="keywordtype">float</span> __nu, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l00649"></a><span class="lineno"> 649</span>  { <span class="keywordflow">return</span> __detail::__cyl_neumann_n<float>(__nu, __x); }</div><div class="line"><a name="l00650"></a><span class="lineno"> 650</span> <span class="comment"></span></div><div class="line"><a name="l00651"></a><span class="lineno"> 651</span> <span class="comment"> /**</span></div><div class="line"><a name="l00652"></a><span class="lineno"> 652</span> <span class="comment"> * Return the Neumann function @f$ N_{\nu}(x) @f$</span></div><div class="line"><a name="l00653"></a><span class="lineno"> 653</span> <span class="comment"> * of <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x @f$.</span></div><div class="line"><a name="l00654"></a><span class="lineno"> 654</span> <span class="comment"> *</span></div><div class="line"><a name="l00655"></a><span class="lineno"> 655</span> <span class="comment"> * @see cyl_neumann for setails.</span></div><div class="line"><a name="l00656"></a><span class="lineno"> 656</span> <span class="comment"> */</span></div><div class="line"><a name="l00657"></a><span class="lineno"> 657</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00658"></a><span class="lineno"><a class="line" href="a01497.html#gaf8986bae9a523c48d861d233835bda8f"> 658</a></span>  <a class="code" href="a01497.html#gaf8986bae9a523c48d861d233835bda8f">cyl_neumannl</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __nu, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l00659"></a><span class="lineno"> 659</span>  { <span class="keywordflow">return</span> __detail::__cyl_neumann_n<long double>(__nu, __x); }</div><div class="line"><a name="l00660"></a><span class="lineno"> 660</span> <span class="comment"></span></div><div class="line"><a name="l00661"></a><span class="lineno"> 661</span> <span class="comment"> /**</span></div><div class="line"><a name="l00662"></a><span class="lineno"> 662</span> <span class="comment"> * Return the Neumann function @f$ N_{\nu}(x) @f$</span></div><div class="line"><a name="l00663"></a><span class="lineno"> 663</span> <span class="comment"> * of real order @f$ \nu @f$ and argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00664"></a><span class="lineno"> 664</span> <span class="comment"> *</span></div><div class="line"><a name="l00665"></a><span class="lineno"> 665</span> <span class="comment"> * The Neumann function is defined by:</span></div><div class="line"><a name="l00666"></a><span class="lineno"> 666</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00667"></a><span class="lineno"> 667</span> <span class="comment"> * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)}</span></div><div class="line"><a name="l00668"></a><span class="lineno"> 668</span> <span class="comment"> * {\sin \nu\pi}</span></div><div class="line"><a name="l00669"></a><span class="lineno"> 669</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00670"></a><span class="lineno"> 670</span> <span class="comment"> * where @f$ x >= 0 @f$ and for integral order @f$ \nu = n @f$</span></div><div class="line"><a name="l00671"></a><span class="lineno"> 671</span> <span class="comment"> * a limit is taken: @f$ lim_{\nu \to n} @f$.</span></div><div class="line"><a name="l00672"></a><span class="lineno"> 672</span> <span class="comment"> *</span></div><div class="line"><a name="l00673"></a><span class="lineno"> 673</span> <span class="comment"> * @tparam _Tpnu The floating-point type of the order @c __nu.</span></div><div class="line"><a name="l00674"></a><span class="lineno"> 674</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l00675"></a><span class="lineno"> 675</span> <span class="comment"> * @param __nu The order</span></div><div class="line"><a name="l00676"></a><span class="lineno"> 676</span> <span class="comment"> * @param __x The argument, <tt> __x >= 0 </tt></span></div><div class="line"><a name="l00677"></a><span class="lineno"> 677</span> <span class="comment"> * @throw std::domain_error if <tt> __x < 0 </tt>.</span></div><div class="line"><a name="l00678"></a><span class="lineno"> 678</span> <span class="comment"> */</span></div><div class="line"><a name="l00679"></a><span class="lineno"> 679</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tpnu, <span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00680"></a><span class="lineno"> 680</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type</div><div class="line"><a name="l00681"></a><span class="lineno"><a class="line" href="a01497.html#ga5b7c72ab85e361cbd73f1a3b5f0725a6"> 681</a></span>  <a class="code" href="a01497.html#ga5b7c72ab85e361cbd73f1a3b5f0725a6">cyl_neumann</a>(_Tpnu __nu, _Tp __x)</div><div class="line"><a name="l00682"></a><span class="lineno"> 682</span>  {</div><div class="line"><a name="l00683"></a><span class="lineno"> 683</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type;</div><div class="line"><a name="l00684"></a><span class="lineno"> 684</span>  <span class="keywordflow">return</span> __detail::__cyl_neumann_n<__type>(__nu, __x);</div><div class="line"><a name="l00685"></a><span class="lineno"> 685</span>  }</div><div class="line"><a name="l00686"></a><span class="lineno"> 686</span> </div><div class="line"><a name="l00687"></a><span class="lineno"> 687</span>  <span class="comment">// Incomplete elliptic integrals of the first kind</span></div><div class="line"><a name="l00688"></a><span class="lineno"> 688</span> <span class="comment"></span></div><div class="line"><a name="l00689"></a><span class="lineno"> 689</span> <span class="comment"> /**</span></div><div class="line"><a name="l00690"></a><span class="lineno"> 690</span> <span class="comment"> * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$</span></div><div class="line"><a name="l00691"></a><span class="lineno"> 691</span> <span class="comment"> * for @c float modulus @f$ k @f$ and angle @f$ \phi @f$.</span></div><div class="line"><a name="l00692"></a><span class="lineno"> 692</span> <span class="comment"> *</span></div><div class="line"><a name="l00693"></a><span class="lineno"> 693</span> <span class="comment"> * @see ellint_1 for details.</span></div><div class="line"><a name="l00694"></a><span class="lineno"> 694</span> <span class="comment"> */</span></div><div class="line"><a name="l00695"></a><span class="lineno"> 695</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00696"></a><span class="lineno"><a class="line" href="a01497.html#ga308d23d70f4b5e848eb7a4173628ef3b"> 696</a></span>  <a class="code" href="a01497.html#ga308d23d70f4b5e848eb7a4173628ef3b">ellint_1f</a>(<span class="keywordtype">float</span> __k, <span class="keywordtype">float</span> __phi)</div><div class="line"><a name="l00697"></a><span class="lineno"> 697</span>  { <span class="keywordflow">return</span> __detail::__ellint_1<float>(__k, __phi); }</div><div class="line"><a name="l00698"></a><span class="lineno"> 698</span> <span class="comment"></span></div><div class="line"><a name="l00699"></a><span class="lineno"> 699</span> <span class="comment"> /**</span></div><div class="line"><a name="l00700"></a><span class="lineno"> 700</span> <span class="comment"> * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$</span></div><div class="line"><a name="l00701"></a><span class="lineno"> 701</span> <span class="comment"> * for <tt>long double</tt> modulus @f$ k @f$ and angle @f$ \phi @f$.</span></div><div class="line"><a name="l00702"></a><span class="lineno"> 702</span> <span class="comment"> *</span></div><div class="line"><a name="l00703"></a><span class="lineno"> 703</span> <span class="comment"> * @see ellint_1 for details.</span></div><div class="line"><a name="l00704"></a><span class="lineno"> 704</span> <span class="comment"> */</span></div><div class="line"><a name="l00705"></a><span class="lineno"> 705</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00706"></a><span class="lineno"><a class="line" href="a01497.html#ga795383fa51e02351000b410b478d824f"> 706</a></span>  <a class="code" href="a01497.html#ga795383fa51e02351000b410b478d824f">ellint_1l</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __k, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __phi)</div><div class="line"><a name="l00707"></a><span class="lineno"> 707</span>  { <span class="keywordflow">return</span> __detail::__ellint_1<long double>(__k, __phi); }</div><div class="line"><a name="l00708"></a><span class="lineno"> 708</span> <span class="comment"></span></div><div class="line"><a name="l00709"></a><span class="lineno"> 709</span> <span class="comment"> /**</span></div><div class="line"><a name="l00710"></a><span class="lineno"> 710</span> <span class="comment"> * Return the incomplete elliptic integral of the first kind @f$ F(k,\phi) @f$</span></div><div class="line"><a name="l00711"></a><span class="lineno"> 711</span> <span class="comment"> * for @c real modulus @f$ k @f$ and angle @f$ \phi @f$.</span></div><div class="line"><a name="l00712"></a><span class="lineno"> 712</span> <span class="comment"> *</span></div><div class="line"><a name="l00713"></a><span class="lineno"> 713</span> <span class="comment"> * The incomplete elliptic integral of the first kind is defined as</span></div><div class="line"><a name="l00714"></a><span class="lineno"> 714</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00715"></a><span class="lineno"> 715</span> <span class="comment"> * F(k,\phi) = \int_0^{\phi}\frac{d\theta}</span></div><div class="line"><a name="l00716"></a><span class="lineno"> 716</span> <span class="comment"> * {\sqrt{1 - k^2 sin^2\theta}}</span></div><div class="line"><a name="l00717"></a><span class="lineno"> 717</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00718"></a><span class="lineno"> 718</span> <span class="comment"> * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of</span></div><div class="line"><a name="l00719"></a><span class="lineno"> 719</span> <span class="comment"> * the first kind, @f$ K(k) @f$. @see comp_ellint_1.</span></div><div class="line"><a name="l00720"></a><span class="lineno"> 720</span> <span class="comment"> *</span></div><div class="line"><a name="l00721"></a><span class="lineno"> 721</span> <span class="comment"> * @tparam _Tp The floating-point type of the modulus @c __k.</span></div><div class="line"><a name="l00722"></a><span class="lineno"> 722</span> <span class="comment"> * @tparam _Tpp The floating-point type of the angle @c __phi.</span></div><div class="line"><a name="l00723"></a><span class="lineno"> 723</span> <span class="comment"> * @param __k The modulus, <tt> abs(__k) <= 1 </tt></span></div><div class="line"><a name="l00724"></a><span class="lineno"> 724</span> <span class="comment"> * @param __phi The integral limit argument in radians</span></div><div class="line"><a name="l00725"></a><span class="lineno"> 725</span> <span class="comment"> * @throw std::domain_error if <tt> abs(__k) > 1 </tt>.</span></div><div class="line"><a name="l00726"></a><span class="lineno"> 726</span> <span class="comment"> */</span></div><div class="line"><a name="l00727"></a><span class="lineno"> 727</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp, <span class="keyword">typename</span> _Tpp></div><div class="line"><a name="l00728"></a><span class="lineno"> 728</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tp, _Tpp>::__type</div><div class="line"><a name="l00729"></a><span class="lineno"><a class="line" href="a01497.html#gae6b3df5556f38a7d72f9b4457d856f9c"> 729</a></span>  <a class="code" href="a01497.html#gae6b3df5556f38a7d72f9b4457d856f9c">ellint_1</a>(_Tp __k, _Tpp __phi)</div><div class="line"><a name="l00730"></a><span class="lineno"> 730</span>  {</div><div class="line"><a name="l00731"></a><span class="lineno"> 731</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tp, _Tpp>::__type __type;</div><div class="line"><a name="l00732"></a><span class="lineno"> 732</span>  <span class="keywordflow">return</span> __detail::__ellint_1<__type>(__k, __phi);</div><div class="line"><a name="l00733"></a><span class="lineno"> 733</span>  }</div><div class="line"><a name="l00734"></a><span class="lineno"> 734</span> </div><div class="line"><a name="l00735"></a><span class="lineno"> 735</span>  <span class="comment">// Incomplete elliptic integrals of the second kind</span></div><div class="line"><a name="l00736"></a><span class="lineno"> 736</span> <span class="comment"></span></div><div class="line"><a name="l00737"></a><span class="lineno"> 737</span> <span class="comment"> /**</span></div><div class="line"><a name="l00738"></a><span class="lineno"> 738</span> <span class="comment"> * @brief Return the incomplete elliptic integral of the second kind</span></div><div class="line"><a name="l00739"></a><span class="lineno"> 739</span> <span class="comment"> * @f$ E(k,\phi) @f$ for @c float argument.</span></div><div class="line"><a name="l00740"></a><span class="lineno"> 740</span> <span class="comment"> *</span></div><div class="line"><a name="l00741"></a><span class="lineno"> 741</span> <span class="comment"> * @see ellint_2 for details.</span></div><div class="line"><a name="l00742"></a><span class="lineno"> 742</span> <span class="comment"> */</span></div><div class="line"><a name="l00743"></a><span class="lineno"> 743</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00744"></a><span class="lineno"><a class="line" href="a01497.html#ga594a730163c6228c75b152462700062b"> 744</a></span>  <a class="code" href="a01497.html#ga594a730163c6228c75b152462700062b">ellint_2f</a>(<span class="keywordtype">float</span> __k, <span class="keywordtype">float</span> __phi)</div><div class="line"><a name="l00745"></a><span class="lineno"> 745</span>  { <span class="keywordflow">return</span> __detail::__ellint_2<float>(__k, __phi); }</div><div class="line"><a name="l00746"></a><span class="lineno"> 746</span> <span class="comment"></span></div><div class="line"><a name="l00747"></a><span class="lineno"> 747</span> <span class="comment"> /**</span></div><div class="line"><a name="l00748"></a><span class="lineno"> 748</span> <span class="comment"> * @brief Return the incomplete elliptic integral of the second kind</span></div><div class="line"><a name="l00749"></a><span class="lineno"> 749</span> <span class="comment"> * @f$ E(k,\phi) @f$.</span></div><div class="line"><a name="l00750"></a><span class="lineno"> 750</span> <span class="comment"> *</span></div><div class="line"><a name="l00751"></a><span class="lineno"> 751</span> <span class="comment"> * @see ellint_2 for details.</span></div><div class="line"><a name="l00752"></a><span class="lineno"> 752</span> <span class="comment"> */</span></div><div class="line"><a name="l00753"></a><span class="lineno"> 753</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00754"></a><span class="lineno"><a class="line" href="a01497.html#ga5c791332d374a809d8ca16c69a1a30f5"> 754</a></span>  <a class="code" href="a01497.html#ga5c791332d374a809d8ca16c69a1a30f5">ellint_2l</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __k, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __phi)</div><div class="line"><a name="l00755"></a><span class="lineno"> 755</span>  { <span class="keywordflow">return</span> __detail::__ellint_2<long double>(__k, __phi); }</div><div class="line"><a name="l00756"></a><span class="lineno"> 756</span> <span class="comment"></span></div><div class="line"><a name="l00757"></a><span class="lineno"> 757</span> <span class="comment"> /**</span></div><div class="line"><a name="l00758"></a><span class="lineno"> 758</span> <span class="comment"> * Return the incomplete elliptic integral of the second kind</span></div><div class="line"><a name="l00759"></a><span class="lineno"> 759</span> <span class="comment"> * @f$ E(k,\phi) @f$.</span></div><div class="line"><a name="l00760"></a><span class="lineno"> 760</span> <span class="comment"> *</span></div><div class="line"><a name="l00761"></a><span class="lineno"> 761</span> <span class="comment"> * The incomplete elliptic integral of the second kind is defined as</span></div><div class="line"><a name="l00762"></a><span class="lineno"> 762</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00763"></a><span class="lineno"> 763</span> <span class="comment"> * E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta}</span></div><div class="line"><a name="l00764"></a><span class="lineno"> 764</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00765"></a><span class="lineno"> 765</span> <span class="comment"> * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of</span></div><div class="line"><a name="l00766"></a><span class="lineno"> 766</span> <span class="comment"> * the second kind, @f$ E(k) @f$. @see comp_ellint_2.</span></div><div class="line"><a name="l00767"></a><span class="lineno"> 767</span> <span class="comment"> *</span></div><div class="line"><a name="l00768"></a><span class="lineno"> 768</span> <span class="comment"> * @tparam _Tp The floating-point type of the modulus @c __k.</span></div><div class="line"><a name="l00769"></a><span class="lineno"> 769</span> <span class="comment"> * @tparam _Tpp The floating-point type of the angle @c __phi.</span></div><div class="line"><a name="l00770"></a><span class="lineno"> 770</span> <span class="comment"> * @param __k The modulus, <tt> abs(__k) <= 1 </tt></span></div><div class="line"><a name="l00771"></a><span class="lineno"> 771</span> <span class="comment"> * @param __phi The integral limit argument in radians</span></div><div class="line"><a name="l00772"></a><span class="lineno"> 772</span> <span class="comment"> * @return The elliptic function of the second kind.</span></div><div class="line"><a name="l00773"></a><span class="lineno"> 773</span> <span class="comment"> * @throw std::domain_error if <tt> abs(__k) > 1 </tt>.</span></div><div class="line"><a name="l00774"></a><span class="lineno"> 774</span> <span class="comment"> */</span></div><div class="line"><a name="l00775"></a><span class="lineno"> 775</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp, <span class="keyword">typename</span> _Tpp></div><div class="line"><a name="l00776"></a><span class="lineno"> 776</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tp, _Tpp>::__type</div><div class="line"><a name="l00777"></a><span class="lineno"><a class="line" href="a01497.html#gad6dd71db2b3f90d24ff49bf8cf37bc37"> 777</a></span>  <a class="code" href="a01497.html#gad6dd71db2b3f90d24ff49bf8cf37bc37">ellint_2</a>(_Tp __k, _Tpp __phi)</div><div class="line"><a name="l00778"></a><span class="lineno"> 778</span>  {</div><div class="line"><a name="l00779"></a><span class="lineno"> 779</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_2<_Tp, _Tpp>::__type __type;</div><div class="line"><a name="l00780"></a><span class="lineno"> 780</span>  <span class="keywordflow">return</span> __detail::__ellint_2<__type>(__k, __phi);</div><div class="line"><a name="l00781"></a><span class="lineno"> 781</span>  }</div><div class="line"><a name="l00782"></a><span class="lineno"> 782</span> </div><div class="line"><a name="l00783"></a><span class="lineno"> 783</span>  <span class="comment">// Incomplete elliptic integrals of the third kind</span></div><div class="line"><a name="l00784"></a><span class="lineno"> 784</span> <span class="comment"></span></div><div class="line"><a name="l00785"></a><span class="lineno"> 785</span> <span class="comment"> /**</span></div><div class="line"><a name="l00786"></a><span class="lineno"> 786</span> <span class="comment"> * @brief Return the incomplete elliptic integral of the third kind</span></div><div class="line"><a name="l00787"></a><span class="lineno"> 787</span> <span class="comment"> * @f$ \Pi(k,\nu,\phi) @f$ for @c float argument.</span></div><div class="line"><a name="l00788"></a><span class="lineno"> 788</span> <span class="comment"> *</span></div><div class="line"><a name="l00789"></a><span class="lineno"> 789</span> <span class="comment"> * @see ellint_3 for details.</span></div><div class="line"><a name="l00790"></a><span class="lineno"> 790</span> <span class="comment"> */</span></div><div class="line"><a name="l00791"></a><span class="lineno"> 791</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00792"></a><span class="lineno"><a class="line" href="a01497.html#ga1a80bd2c15bc9fbecda2630a9e9409e7"> 792</a></span>  <a class="code" href="a01497.html#ga1a80bd2c15bc9fbecda2630a9e9409e7">ellint_3f</a>(<span class="keywordtype">float</span> __k, <span class="keywordtype">float</span> __nu, <span class="keywordtype">float</span> __phi)</div><div class="line"><a name="l00793"></a><span class="lineno"> 793</span>  { <span class="keywordflow">return</span> __detail::__ellint_3<float>(__k, __nu, __phi); }</div><div class="line"><a name="l00794"></a><span class="lineno"> 794</span> <span class="comment"></span></div><div class="line"><a name="l00795"></a><span class="lineno"> 795</span> <span class="comment"> /**</span></div><div class="line"><a name="l00796"></a><span class="lineno"> 796</span> <span class="comment"> * @brief Return the incomplete elliptic integral of the third kind</span></div><div class="line"><a name="l00797"></a><span class="lineno"> 797</span> <span class="comment"> * @f$ \Pi(k,\nu,\phi) @f$.</span></div><div class="line"><a name="l00798"></a><span class="lineno"> 798</span> <span class="comment"> *</span></div><div class="line"><a name="l00799"></a><span class="lineno"> 799</span> <span class="comment"> * @see ellint_3 for details.</span></div><div class="line"><a name="l00800"></a><span class="lineno"> 800</span> <span class="comment"> */</span></div><div class="line"><a name="l00801"></a><span class="lineno"> 801</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00802"></a><span class="lineno"><a class="line" href="a01497.html#gaa8c0e5864df8769021a7f3e21a30c5d2"> 802</a></span>  <a class="code" href="a01497.html#gaa8c0e5864df8769021a7f3e21a30c5d2">ellint_3l</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __k, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __nu, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __phi)</div><div class="line"><a name="l00803"></a><span class="lineno"> 803</span>  { <span class="keywordflow">return</span> __detail::__ellint_3<long double>(__k, __nu, __phi); }</div><div class="line"><a name="l00804"></a><span class="lineno"> 804</span> <span class="comment"></span></div><div class="line"><a name="l00805"></a><span class="lineno"> 805</span> <span class="comment"> /**</span></div><div class="line"><a name="l00806"></a><span class="lineno"> 806</span> <span class="comment"> * @brief Return the incomplete elliptic integral of the third kind</span></div><div class="line"><a name="l00807"></a><span class="lineno"> 807</span> <span class="comment"> * @f$ \Pi(k,\nu,\phi) @f$.</span></div><div class="line"><a name="l00808"></a><span class="lineno"> 808</span> <span class="comment"> *</span></div><div class="line"><a name="l00809"></a><span class="lineno"> 809</span> <span class="comment"> * The incomplete elliptic integral of the third kind is defined by:</span></div><div class="line"><a name="l00810"></a><span class="lineno"> 810</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00811"></a><span class="lineno"> 811</span> <span class="comment"> * \Pi(k,\nu,\phi) = \int_0^{\phi}</span></div><div class="line"><a name="l00812"></a><span class="lineno"> 812</span> <span class="comment"> * \frac{d\theta}</span></div><div class="line"><a name="l00813"></a><span class="lineno"> 813</span> <span class="comment"> * {(1 - \nu \sin^2\theta)</span></div><div class="line"><a name="l00814"></a><span class="lineno"> 814</span> <span class="comment"> * \sqrt{1 - k^2 \sin^2\theta}}</span></div><div class="line"><a name="l00815"></a><span class="lineno"> 815</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00816"></a><span class="lineno"> 816</span> <span class="comment"> * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of</span></div><div class="line"><a name="l00817"></a><span class="lineno"> 817</span> <span class="comment"> * the third kind, @f$ \Pi(k,\nu) @f$. @see comp_ellint_3.</span></div><div class="line"><a name="l00818"></a><span class="lineno"> 818</span> <span class="comment"> *</span></div><div class="line"><a name="l00819"></a><span class="lineno"> 819</span> <span class="comment"> * @tparam _Tp The floating-point type of the modulus @c __k.</span></div><div class="line"><a name="l00820"></a><span class="lineno"> 820</span> <span class="comment"> * @tparam _Tpn The floating-point type of the argument @c __nu.</span></div><div class="line"><a name="l00821"></a><span class="lineno"> 821</span> <span class="comment"> * @tparam _Tpp The floating-point type of the angle @c __phi.</span></div><div class="line"><a name="l00822"></a><span class="lineno"> 822</span> <span class="comment"> * @param __k The modulus, <tt> abs(__k) <= 1 </tt></span></div><div class="line"><a name="l00823"></a><span class="lineno"> 823</span> <span class="comment"> * @param __nu The second argument</span></div><div class="line"><a name="l00824"></a><span class="lineno"> 824</span> <span class="comment"> * @param __phi The integral limit argument in radians</span></div><div class="line"><a name="l00825"></a><span class="lineno"> 825</span> <span class="comment"> * @return The elliptic function of the third kind.</span></div><div class="line"><a name="l00826"></a><span class="lineno"> 826</span> <span class="comment"> * @throw std::domain_error if <tt> abs(__k) > 1 </tt>.</span></div><div class="line"><a name="l00827"></a><span class="lineno"> 827</span> <span class="comment"> */</span></div><div class="line"><a name="l00828"></a><span class="lineno"> 828</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp, <span class="keyword">typename</span> _Tpn, <span class="keyword">typename</span> _Tpp></div><div class="line"><a name="l00829"></a><span class="lineno"> 829</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type</div><div class="line"><a name="l00830"></a><span class="lineno"><a class="line" href="a01497.html#ga20832e3a67d25cc3d415cafc88019ac3"> 830</a></span>  <a class="code" href="a01497.html#ga20832e3a67d25cc3d415cafc88019ac3">ellint_3</a>(_Tp __k, _Tpn __nu, _Tpp __phi)</div><div class="line"><a name="l00831"></a><span class="lineno"> 831</span>  {</div><div class="line"><a name="l00832"></a><span class="lineno"> 832</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type __type;</div><div class="line"><a name="l00833"></a><span class="lineno"> 833</span>  <span class="keywordflow">return</span> __detail::__ellint_3<__type>(__k, __nu, __phi);</div><div class="line"><a name="l00834"></a><span class="lineno"> 834</span>  }</div><div class="line"><a name="l00835"></a><span class="lineno"> 835</span> </div><div class="line"><a name="l00836"></a><span class="lineno"> 836</span>  <span class="comment">// Exponential integrals</span></div><div class="line"><a name="l00837"></a><span class="lineno"> 837</span> <span class="comment"></span></div><div class="line"><a name="l00838"></a><span class="lineno"> 838</span> <span class="comment"> /**</span></div><div class="line"><a name="l00839"></a><span class="lineno"> 839</span> <span class="comment"> * Return the exponential integral @f$ Ei(x) @f$ for @c float argument @c x.</span></div><div class="line"><a name="l00840"></a><span class="lineno"> 840</span> <span class="comment"> *</span></div><div class="line"><a name="l00841"></a><span class="lineno"> 841</span> <span class="comment"> * @see expint for details.</span></div><div class="line"><a name="l00842"></a><span class="lineno"> 842</span> <span class="comment"> */</span></div><div class="line"><a name="l00843"></a><span class="lineno"> 843</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00844"></a><span class="lineno"><a class="line" href="a01497.html#ga5842816f6eed2594e0a327df4e4a2a47"> 844</a></span>  <a class="code" href="a01497.html#ga5842816f6eed2594e0a327df4e4a2a47">expintf</a>(<span class="keywordtype">float</span> __x)</div><div class="line"><a name="l00845"></a><span class="lineno"> 845</span>  { <span class="keywordflow">return</span> __detail::__expint<float>(__x); }</div><div class="line"><a name="l00846"></a><span class="lineno"> 846</span> <span class="comment"></span></div><div class="line"><a name="l00847"></a><span class="lineno"> 847</span> <span class="comment"> /**</span></div><div class="line"><a name="l00848"></a><span class="lineno"> 848</span> <span class="comment"> * Return the exponential integral @f$ Ei(x) @f$</span></div><div class="line"><a name="l00849"></a><span class="lineno"> 849</span> <span class="comment"> * for <tt>long double</tt> argument @c x.</span></div><div class="line"><a name="l00850"></a><span class="lineno"> 850</span> <span class="comment"> *</span></div><div class="line"><a name="l00851"></a><span class="lineno"> 851</span> <span class="comment"> * @see expint for details.</span></div><div class="line"><a name="l00852"></a><span class="lineno"> 852</span> <span class="comment"> */</span></div><div class="line"><a name="l00853"></a><span class="lineno"> 853</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00854"></a><span class="lineno"><a class="line" href="a01497.html#ga1329130b32328d0666e290ee5931fa4f"> 854</a></span>  <a class="code" href="a01497.html#ga1329130b32328d0666e290ee5931fa4f">expintl</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l00855"></a><span class="lineno"> 855</span>  { <span class="keywordflow">return</span> __detail::__expint<long double>(__x); }</div><div class="line"><a name="l00856"></a><span class="lineno"> 856</span> <span class="comment"></span></div><div class="line"><a name="l00857"></a><span class="lineno"> 857</span> <span class="comment"> /**</span></div><div class="line"><a name="l00858"></a><span class="lineno"> 858</span> <span class="comment"> * Return the exponential integral @f$ Ei(x) @f$ for @c real argument @c x.</span></div><div class="line"><a name="l00859"></a><span class="lineno"> 859</span> <span class="comment"> *</span></div><div class="line"><a name="l00860"></a><span class="lineno"> 860</span> <span class="comment"> * The exponential integral is given by</span></div><div class="line"><a name="l00861"></a><span class="lineno"> 861</span> <span class="comment"> * \f[</span></div><div class="line"><a name="l00862"></a><span class="lineno"> 862</span> <span class="comment"> * Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt</span></div><div class="line"><a name="l00863"></a><span class="lineno"> 863</span> <span class="comment"> * \f]</span></div><div class="line"><a name="l00864"></a><span class="lineno"> 864</span> <span class="comment"> *</span></div><div class="line"><a name="l00865"></a><span class="lineno"> 865</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l00866"></a><span class="lineno"> 866</span> <span class="comment"> * @param __x The argument of the exponential integral function.</span></div><div class="line"><a name="l00867"></a><span class="lineno"> 867</span> <span class="comment"> */</span></div><div class="line"><a name="l00868"></a><span class="lineno"> 868</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00869"></a><span class="lineno"> 869</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l00870"></a><span class="lineno"><a class="line" href="a01497.html#ga88ba17f5d050a6973ca4db1bf6e90590"> 870</a></span>  <a class="code" href="a01497.html#ga88ba17f5d050a6973ca4db1bf6e90590">expint</a>(_Tp __x)</div><div class="line"><a name="l00871"></a><span class="lineno"> 871</span>  {</div><div class="line"><a name="l00872"></a><span class="lineno"> 872</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l00873"></a><span class="lineno"> 873</span>  <span class="keywordflow">return</span> __detail::__expint<__type>(__x);</div><div class="line"><a name="l00874"></a><span class="lineno"> 874</span>  }</div><div class="line"><a name="l00875"></a><span class="lineno"> 875</span> </div><div class="line"><a name="l00876"></a><span class="lineno"> 876</span>  <span class="comment">// Hermite polynomials</span></div><div class="line"><a name="l00877"></a><span class="lineno"> 877</span> <span class="comment"></span></div><div class="line"><a name="l00878"></a><span class="lineno"> 878</span> <span class="comment"> /**</span></div><div class="line"><a name="l00879"></a><span class="lineno"> 879</span> <span class="comment"> * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n</span></div><div class="line"><a name="l00880"></a><span class="lineno"> 880</span> <span class="comment"> * and float argument @c x.</span></div><div class="line"><a name="l00881"></a><span class="lineno"> 881</span> <span class="comment"> *</span></div><div class="line"><a name="l00882"></a><span class="lineno"> 882</span> <span class="comment"> * @see hermite for details.</span></div><div class="line"><a name="l00883"></a><span class="lineno"> 883</span> <span class="comment"> */</span></div><div class="line"><a name="l00884"></a><span class="lineno"> 884</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00885"></a><span class="lineno"><a class="line" href="a01497.html#ga94dae7444bb349e33057a92932db8abe"> 885</a></span>  <a class="code" href="a01497.html#ga94dae7444bb349e33057a92932db8abe">hermitef</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l00886"></a><span class="lineno"> 886</span>  { <span class="keywordflow">return</span> __detail::__poly_hermite<float>(__n, __x); }</div><div class="line"><a name="l00887"></a><span class="lineno"> 887</span> <span class="comment"></span></div><div class="line"><a name="l00888"></a><span class="lineno"> 888</span> <span class="comment"> /**</span></div><div class="line"><a name="l00889"></a><span class="lineno"> 889</span> <span class="comment"> * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n</span></div><div class="line"><a name="l00890"></a><span class="lineno"> 890</span> <span class="comment"> * and <tt>long double</tt> argument @c x.</span></div><div class="line"><a name="l00891"></a><span class="lineno"> 891</span> <span class="comment"> *</span></div><div class="line"><a name="l00892"></a><span class="lineno"> 892</span> <span class="comment"> * @see hermite for details.</span></div><div class="line"><a name="l00893"></a><span class="lineno"> 893</span> <span class="comment"> */</span></div><div class="line"><a name="l00894"></a><span class="lineno"> 894</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00895"></a><span class="lineno"><a class="line" href="a01497.html#ga21f8e312ee3e65286f86bf141b0f32e0"> 895</a></span>  <a class="code" href="a01497.html#ga21f8e312ee3e65286f86bf141b0f32e0">hermitel</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l00896"></a><span class="lineno"> 896</span>  { <span class="keywordflow">return</span> __detail::__poly_hermite<long double>(__n, __x); }</div><div class="line"><a name="l00897"></a><span class="lineno"> 897</span> <span class="comment"></span></div><div class="line"><a name="l00898"></a><span class="lineno"> 898</span> <span class="comment"> /**</span></div><div class="line"><a name="l00899"></a><span class="lineno"> 899</span> <span class="comment"> * Return the Hermite polynomial @f$ H_n(x) @f$ of order n</span></div><div class="line"><a name="l00900"></a><span class="lineno"> 900</span> <span class="comment"> * and @c real argument @c x.</span></div><div class="line"><a name="l00901"></a><span class="lineno"> 901</span> <span class="comment"> *</span></div><div class="line"><a name="l00902"></a><span class="lineno"> 902</span> <span class="comment"> * The Hermite polynomial is defined by:</span></div><div class="line"><a name="l00903"></a><span class="lineno"> 903</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00904"></a><span class="lineno"> 904</span> <span class="comment"> * H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}</span></div><div class="line"><a name="l00905"></a><span class="lineno"> 905</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00906"></a><span class="lineno"> 906</span> <span class="comment"> *</span></div><div class="line"><a name="l00907"></a><span class="lineno"> 907</span> <span class="comment"> * The Hermite polynomial obeys a reflection formula:</span></div><div class="line"><a name="l00908"></a><span class="lineno"> 908</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00909"></a><span class="lineno"> 909</span> <span class="comment"> * H_n(-x) = (-1)^n H_n(x)</span></div><div class="line"><a name="l00910"></a><span class="lineno"> 910</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00911"></a><span class="lineno"> 911</span> <span class="comment"> *</span></div><div class="line"><a name="l00912"></a><span class="lineno"> 912</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l00913"></a><span class="lineno"> 913</span> <span class="comment"> * @param __n The order</span></div><div class="line"><a name="l00914"></a><span class="lineno"> 914</span> <span class="comment"> * @param __x The argument</span></div><div class="line"><a name="l00915"></a><span class="lineno"> 915</span> <span class="comment"> */</span></div><div class="line"><a name="l00916"></a><span class="lineno"> 916</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00917"></a><span class="lineno"> 917</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l00918"></a><span class="lineno"><a class="line" href="a01497.html#ga97632b8bf77c323b2369e8d327965bdf"> 918</a></span>  <a class="code" href="a01497.html#ga97632b8bf77c323b2369e8d327965bdf">hermite</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, _Tp __x)</div><div class="line"><a name="l00919"></a><span class="lineno"> 919</span>  {</div><div class="line"><a name="l00920"></a><span class="lineno"> 920</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l00921"></a><span class="lineno"> 921</span>  <span class="keywordflow">return</span> __detail::__poly_hermite<__type>(__n, __x);</div><div class="line"><a name="l00922"></a><span class="lineno"> 922</span>  }</div><div class="line"><a name="l00923"></a><span class="lineno"> 923</span> </div><div class="line"><a name="l00924"></a><span class="lineno"> 924</span>  <span class="comment">// Laguerre polynomials</span></div><div class="line"><a name="l00925"></a><span class="lineno"> 925</span> <span class="comment"></span></div><div class="line"><a name="l00926"></a><span class="lineno"> 926</span> <span class="comment"> /**</span></div><div class="line"><a name="l00927"></a><span class="lineno"> 927</span> <span class="comment"> * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n</span></div><div class="line"><a name="l00928"></a><span class="lineno"> 928</span> <span class="comment"> * and @c float argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00929"></a><span class="lineno"> 929</span> <span class="comment"> *</span></div><div class="line"><a name="l00930"></a><span class="lineno"> 930</span> <span class="comment"> * @see laguerre for more details.</span></div><div class="line"><a name="l00931"></a><span class="lineno"> 931</span> <span class="comment"> */</span></div><div class="line"><a name="l00932"></a><span class="lineno"> 932</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00933"></a><span class="lineno"><a class="line" href="a01497.html#gada763419b0e21b38e38daa8b6eb56a8c"> 933</a></span>  <a class="code" href="a01497.html#gada763419b0e21b38e38daa8b6eb56a8c">laguerref</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l00934"></a><span class="lineno"> 934</span>  { <span class="keywordflow">return</span> __detail::__laguerre<float>(__n, __x); }</div><div class="line"><a name="l00935"></a><span class="lineno"> 935</span> <span class="comment"></span></div><div class="line"><a name="l00936"></a><span class="lineno"> 936</span> <span class="comment"> /**</span></div><div class="line"><a name="l00937"></a><span class="lineno"> 937</span> <span class="comment"> * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n</span></div><div class="line"><a name="l00938"></a><span class="lineno"> 938</span> <span class="comment"> * and <tt>long double</tt> argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00939"></a><span class="lineno"> 939</span> <span class="comment"> *</span></div><div class="line"><a name="l00940"></a><span class="lineno"> 940</span> <span class="comment"> * @see laguerre for more details.</span></div><div class="line"><a name="l00941"></a><span class="lineno"> 941</span> <span class="comment"> */</span></div><div class="line"><a name="l00942"></a><span class="lineno"> 942</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00943"></a><span class="lineno"><a class="line" href="a01497.html#gaaf8b141edf9163b37ea4f2ed3e0191fc"> 943</a></span>  <a class="code" href="a01497.html#gaaf8b141edf9163b37ea4f2ed3e0191fc">laguerrel</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l00944"></a><span class="lineno"> 944</span>  { <span class="keywordflow">return</span> __detail::__laguerre<long double>(__n, __x); }</div><div class="line"><a name="l00945"></a><span class="lineno"> 945</span> <span class="comment"></span></div><div class="line"><a name="l00946"></a><span class="lineno"> 946</span> <span class="comment"> /**</span></div><div class="line"><a name="l00947"></a><span class="lineno"> 947</span> <span class="comment"> * Returns the Laguerre polynomial @f$ L_n(x) @f$</span></div><div class="line"><a name="l00948"></a><span class="lineno"> 948</span> <span class="comment"> * of nonnegative degree @c n and real argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l00949"></a><span class="lineno"> 949</span> <span class="comment"> *</span></div><div class="line"><a name="l00950"></a><span class="lineno"> 950</span> <span class="comment"> * The Laguerre polynomial is defined by:</span></div><div class="line"><a name="l00951"></a><span class="lineno"> 951</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00952"></a><span class="lineno"> 952</span> <span class="comment"> * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})</span></div><div class="line"><a name="l00953"></a><span class="lineno"> 953</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00954"></a><span class="lineno"> 954</span> <span class="comment"> *</span></div><div class="line"><a name="l00955"></a><span class="lineno"> 955</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l00956"></a><span class="lineno"> 956</span> <span class="comment"> * @param __n The nonnegative order</span></div><div class="line"><a name="l00957"></a><span class="lineno"> 957</span> <span class="comment"> * @param __x The argument <tt> __x >= 0 </tt></span></div><div class="line"><a name="l00958"></a><span class="lineno"> 958</span> <span class="comment"> * @throw std::domain_error if <tt> __x < 0 </tt>.</span></div><div class="line"><a name="l00959"></a><span class="lineno"> 959</span> <span class="comment"> */</span></div><div class="line"><a name="l00960"></a><span class="lineno"> 960</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l00961"></a><span class="lineno"> 961</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l00962"></a><span class="lineno"><a class="line" href="a01497.html#gacae65579b397fddcd27954380d364a58"> 962</a></span>  <a class="code" href="a01497.html#gacae65579b397fddcd27954380d364a58">laguerre</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, _Tp __x)</div><div class="line"><a name="l00963"></a><span class="lineno"> 963</span>  {</div><div class="line"><a name="l00964"></a><span class="lineno"> 964</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l00965"></a><span class="lineno"> 965</span>  <span class="keywordflow">return</span> __detail::__laguerre<__type>(__n, __x);</div><div class="line"><a name="l00966"></a><span class="lineno"> 966</span>  }</div><div class="line"><a name="l00967"></a><span class="lineno"> 967</span> </div><div class="line"><a name="l00968"></a><span class="lineno"> 968</span>  <span class="comment">// Legendre polynomials</span></div><div class="line"><a name="l00969"></a><span class="lineno"> 969</span> <span class="comment"></span></div><div class="line"><a name="l00970"></a><span class="lineno"> 970</span> <span class="comment"> /**</span></div><div class="line"><a name="l00971"></a><span class="lineno"> 971</span> <span class="comment"> * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative</span></div><div class="line"><a name="l00972"></a><span class="lineno"> 972</span> <span class="comment"> * degree @f$ l @f$ and @c float argument @f$ |x| <= 0 @f$.</span></div><div class="line"><a name="l00973"></a><span class="lineno"> 973</span> <span class="comment"> *</span></div><div class="line"><a name="l00974"></a><span class="lineno"> 974</span> <span class="comment"> * @see legendre for more details.</span></div><div class="line"><a name="l00975"></a><span class="lineno"> 975</span> <span class="comment"> */</span></div><div class="line"><a name="l00976"></a><span class="lineno"> 976</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l00977"></a><span class="lineno"><a class="line" href="a01497.html#gaed94e3c664c99f5204da75be75aeac21"> 977</a></span>  <a class="code" href="a01497.html#gaed94e3c664c99f5204da75be75aeac21">legendref</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __l, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l00978"></a><span class="lineno"> 978</span>  { <span class="keywordflow">return</span> __detail::__poly_legendre_p<float>(__l, __x); }</div><div class="line"><a name="l00979"></a><span class="lineno"> 979</span> <span class="comment"></span></div><div class="line"><a name="l00980"></a><span class="lineno"> 980</span> <span class="comment"> /**</span></div><div class="line"><a name="l00981"></a><span class="lineno"> 981</span> <span class="comment"> * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative</span></div><div class="line"><a name="l00982"></a><span class="lineno"> 982</span> <span class="comment"> * degree @f$ l @f$ and <tt>long double</tt> argument @f$ |x| <= 0 @f$.</span></div><div class="line"><a name="l00983"></a><span class="lineno"> 983</span> <span class="comment"> *</span></div><div class="line"><a name="l00984"></a><span class="lineno"> 984</span> <span class="comment"> * @see legendre for more details.</span></div><div class="line"><a name="l00985"></a><span class="lineno"> 985</span> <span class="comment"> */</span></div><div class="line"><a name="l00986"></a><span class="lineno"> 986</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l00987"></a><span class="lineno"><a class="line" href="a01497.html#ga1b39bc22e3cc4860d08eb54099460391"> 987</a></span>  <a class="code" href="a01497.html#ga1b39bc22e3cc4860d08eb54099460391">legendrel</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __l, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l00988"></a><span class="lineno"> 988</span>  { <span class="keywordflow">return</span> __detail::__poly_legendre_p<long double>(__l, __x); }</div><div class="line"><a name="l00989"></a><span class="lineno"> 989</span> <span class="comment"></span></div><div class="line"><a name="l00990"></a><span class="lineno"> 990</span> <span class="comment"> /**</span></div><div class="line"><a name="l00991"></a><span class="lineno"> 991</span> <span class="comment"> * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative</span></div><div class="line"><a name="l00992"></a><span class="lineno"> 992</span> <span class="comment"> * degree @f$ l @f$ and real argument @f$ |x| <= 0 @f$.</span></div><div class="line"><a name="l00993"></a><span class="lineno"> 993</span> <span class="comment"> *</span></div><div class="line"><a name="l00994"></a><span class="lineno"> 994</span> <span class="comment"> * The Legendre function of order @f$ l @f$ and argument @f$ x @f$,</span></div><div class="line"><a name="l00995"></a><span class="lineno"> 995</span> <span class="comment"> * @f$ P_l(x) @f$, is defined by:</span></div><div class="line"><a name="l00996"></a><span class="lineno"> 996</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l00997"></a><span class="lineno"> 997</span> <span class="comment"> * P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l}</span></div><div class="line"><a name="l00998"></a><span class="lineno"> 998</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l00999"></a><span class="lineno"> 999</span> <span class="comment"> *</span></div><div class="line"><a name="l01000"></a><span class="lineno"> 1000</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l01001"></a><span class="lineno"> 1001</span> <span class="comment"> * @param __l The degree @f$ l >= 0 @f$</span></div><div class="line"><a name="l01002"></a><span class="lineno"> 1002</span> <span class="comment"> * @param __x The argument @c abs(__x) <= 1</span></div><div class="line"><a name="l01003"></a><span class="lineno"> 1003</span> <span class="comment"> * @throw std::domain_error if @c abs(__x) > 1</span></div><div class="line"><a name="l01004"></a><span class="lineno"> 1004</span> <span class="comment"> */</span></div><div class="line"><a name="l01005"></a><span class="lineno"> 1005</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l01006"></a><span class="lineno"> 1006</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l01007"></a><span class="lineno"><a class="line" href="a01497.html#gaf6eac7fcb98e25b8f3f7d1b95fa7add8"> 1007</a></span>  <a class="code" href="a01497.html#gaf6eac7fcb98e25b8f3f7d1b95fa7add8">legendre</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __l, _Tp __x)</div><div class="line"><a name="l01008"></a><span class="lineno"> 1008</span>  {</div><div class="line"><a name="l01009"></a><span class="lineno"> 1009</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l01010"></a><span class="lineno"> 1010</span>  <span class="keywordflow">return</span> __detail::__poly_legendre_p<__type>(__l, __x);</div><div class="line"><a name="l01011"></a><span class="lineno"> 1011</span>  }</div><div class="line"><a name="l01012"></a><span class="lineno"> 1012</span> </div><div class="line"><a name="l01013"></a><span class="lineno"> 1013</span>  <span class="comment">// Riemann zeta functions</span></div><div class="line"><a name="l01014"></a><span class="lineno"> 1014</span> <span class="comment"></span></div><div class="line"><a name="l01015"></a><span class="lineno"> 1015</span> <span class="comment"> /**</span></div><div class="line"><a name="l01016"></a><span class="lineno"> 1016</span> <span class="comment"> * Return the Riemann zeta function @f$ \zeta(s) @f$</span></div><div class="line"><a name="l01017"></a><span class="lineno"> 1017</span> <span class="comment"> * for @c float argument @f$ s @f$.</span></div><div class="line"><a name="l01018"></a><span class="lineno"> 1018</span> <span class="comment"> *</span></div><div class="line"><a name="l01019"></a><span class="lineno"> 1019</span> <span class="comment"> * @see riemann_zeta for more details.</span></div><div class="line"><a name="l01020"></a><span class="lineno"> 1020</span> <span class="comment"> */</span></div><div class="line"><a name="l01021"></a><span class="lineno"> 1021</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l01022"></a><span class="lineno"><a class="line" href="a01497.html#gaf92063315061a56d3e2c4053156d968e"> 1022</a></span>  <a class="code" href="a01497.html#gaf92063315061a56d3e2c4053156d968e">riemann_zetaf</a>(<span class="keywordtype">float</span> __s)</div><div class="line"><a name="l01023"></a><span class="lineno"> 1023</span>  { <span class="keywordflow">return</span> __detail::__riemann_zeta<float>(__s); }</div><div class="line"><a name="l01024"></a><span class="lineno"> 1024</span> <span class="comment"></span></div><div class="line"><a name="l01025"></a><span class="lineno"> 1025</span> <span class="comment"> /**</span></div><div class="line"><a name="l01026"></a><span class="lineno"> 1026</span> <span class="comment"> * Return the Riemann zeta function @f$ \zeta(s) @f$</span></div><div class="line"><a name="l01027"></a><span class="lineno"> 1027</span> <span class="comment"> * for <tt>long double</tt> argument @f$ s @f$.</span></div><div class="line"><a name="l01028"></a><span class="lineno"> 1028</span> <span class="comment"> *</span></div><div class="line"><a name="l01029"></a><span class="lineno"> 1029</span> <span class="comment"> * @see riemann_zeta for more details.</span></div><div class="line"><a name="l01030"></a><span class="lineno"> 1030</span> <span class="comment"> */</span></div><div class="line"><a name="l01031"></a><span class="lineno"> 1031</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l01032"></a><span class="lineno"><a class="line" href="a01497.html#ga1e92da3b878d75270f38d3ec9b513086"> 1032</a></span>  <a class="code" href="a01497.html#ga1e92da3b878d75270f38d3ec9b513086">riemann_zetal</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __s)</div><div class="line"><a name="l01033"></a><span class="lineno"> 1033</span>  { <span class="keywordflow">return</span> __detail::__riemann_zeta<long double>(__s); }</div><div class="line"><a name="l01034"></a><span class="lineno"> 1034</span> <span class="comment"></span></div><div class="line"><a name="l01035"></a><span class="lineno"> 1035</span> <span class="comment"> /**</span></div><div class="line"><a name="l01036"></a><span class="lineno"> 1036</span> <span class="comment"> * Return the Riemann zeta function @f$ \zeta(s) @f$</span></div><div class="line"><a name="l01037"></a><span class="lineno"> 1037</span> <span class="comment"> * for real argument @f$ s @f$.</span></div><div class="line"><a name="l01038"></a><span class="lineno"> 1038</span> <span class="comment"> *</span></div><div class="line"><a name="l01039"></a><span class="lineno"> 1039</span> <span class="comment"> * The Riemann zeta function is defined by:</span></div><div class="line"><a name="l01040"></a><span class="lineno"> 1040</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l01041"></a><span class="lineno"> 1041</span> <span class="comment"> * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1</span></div><div class="line"><a name="l01042"></a><span class="lineno"> 1042</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l01043"></a><span class="lineno"> 1043</span> <span class="comment"> * and</span></div><div class="line"><a name="l01044"></a><span class="lineno"> 1044</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l01045"></a><span class="lineno"> 1045</span> <span class="comment"> * \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s}</span></div><div class="line"><a name="l01046"></a><span class="lineno"> 1046</span> <span class="comment"> * \hbox{ for } 0 <= s <= 1</span></div><div class="line"><a name="l01047"></a><span class="lineno"> 1047</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l01048"></a><span class="lineno"> 1048</span> <span class="comment"> * For s < 1 use the reflection formula:</span></div><div class="line"><a name="l01049"></a><span class="lineno"> 1049</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l01050"></a><span class="lineno"> 1050</span> <span class="comment"> * \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)</span></div><div class="line"><a name="l01051"></a><span class="lineno"> 1051</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l01052"></a><span class="lineno"> 1052</span> <span class="comment"> *</span></div><div class="line"><a name="l01053"></a><span class="lineno"> 1053</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __s.</span></div><div class="line"><a name="l01054"></a><span class="lineno"> 1054</span> <span class="comment"> * @param __s The argument <tt> s != 1 </tt></span></div><div class="line"><a name="l01055"></a><span class="lineno"> 1055</span> <span class="comment"> */</span></div><div class="line"><a name="l01056"></a><span class="lineno"> 1056</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l01057"></a><span class="lineno"> 1057</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l01058"></a><span class="lineno"><a class="line" href="a01497.html#ga67a6bfed9b6ab692e8c798b674431424"> 1058</a></span>  <a class="code" href="a01497.html#ga67a6bfed9b6ab692e8c798b674431424">riemann_zeta</a>(_Tp __s)</div><div class="line"><a name="l01059"></a><span class="lineno"> 1059</span>  {</div><div class="line"><a name="l01060"></a><span class="lineno"> 1060</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l01061"></a><span class="lineno"> 1061</span>  <span class="keywordflow">return</span> __detail::__riemann_zeta<__type>(__s);</div><div class="line"><a name="l01062"></a><span class="lineno"> 1062</span>  }</div><div class="line"><a name="l01063"></a><span class="lineno"> 1063</span> </div><div class="line"><a name="l01064"></a><span class="lineno"> 1064</span>  <span class="comment">// Spherical Bessel functions</span></div><div class="line"><a name="l01065"></a><span class="lineno"> 1065</span> <span class="comment"></span></div><div class="line"><a name="l01066"></a><span class="lineno"> 1066</span> <span class="comment"> /**</span></div><div class="line"><a name="l01067"></a><span class="lineno"> 1067</span> <span class="comment"> * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n</span></div><div class="line"><a name="l01068"></a><span class="lineno"> 1068</span> <span class="comment"> * and @c float argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l01069"></a><span class="lineno"> 1069</span> <span class="comment"> *</span></div><div class="line"><a name="l01070"></a><span class="lineno"> 1070</span> <span class="comment"> * @see sph_bessel for more details.</span></div><div class="line"><a name="l01071"></a><span class="lineno"> 1071</span> <span class="comment"> */</span></div><div class="line"><a name="l01072"></a><span class="lineno"> 1072</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l01073"></a><span class="lineno"><a class="line" href="a01497.html#ga534e36e1dcefad8daec98920db16eec4"> 1073</a></span>  <a class="code" href="a01497.html#ga534e36e1dcefad8daec98920db16eec4">sph_besself</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l01074"></a><span class="lineno"> 1074</span>  { <span class="keywordflow">return</span> __detail::__sph_bessel<float>(__n, __x); }</div><div class="line"><a name="l01075"></a><span class="lineno"> 1075</span> <span class="comment"></span></div><div class="line"><a name="l01076"></a><span class="lineno"> 1076</span> <span class="comment"> /**</span></div><div class="line"><a name="l01077"></a><span class="lineno"> 1077</span> <span class="comment"> * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n</span></div><div class="line"><a name="l01078"></a><span class="lineno"> 1078</span> <span class="comment"> * and <tt>long double</tt> argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l01079"></a><span class="lineno"> 1079</span> <span class="comment"> *</span></div><div class="line"><a name="l01080"></a><span class="lineno"> 1080</span> <span class="comment"> * @see sph_bessel for more details.</span></div><div class="line"><a name="l01081"></a><span class="lineno"> 1081</span> <span class="comment"> */</span></div><div class="line"><a name="l01082"></a><span class="lineno"> 1082</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l01083"></a><span class="lineno"><a class="line" href="a01497.html#ga11d72b1af81ce9da3c878a25087ee927"> 1083</a></span>  <a class="code" href="a01497.html#ga11d72b1af81ce9da3c878a25087ee927">sph_bessell</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l01084"></a><span class="lineno"> 1084</span>  { <span class="keywordflow">return</span> __detail::__sph_bessel<long double>(__n, __x); }</div><div class="line"><a name="l01085"></a><span class="lineno"> 1085</span> <span class="comment"></span></div><div class="line"><a name="l01086"></a><span class="lineno"> 1086</span> <span class="comment"> /**</span></div><div class="line"><a name="l01087"></a><span class="lineno"> 1087</span> <span class="comment"> * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n</span></div><div class="line"><a name="l01088"></a><span class="lineno"> 1088</span> <span class="comment"> * and real argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l01089"></a><span class="lineno"> 1089</span> <span class="comment"> *</span></div><div class="line"><a name="l01090"></a><span class="lineno"> 1090</span> <span class="comment"> * The spherical Bessel function is defined by:</span></div><div class="line"><a name="l01091"></a><span class="lineno"> 1091</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l01092"></a><span class="lineno"> 1092</span> <span class="comment"> * j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x)</span></div><div class="line"><a name="l01093"></a><span class="lineno"> 1093</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l01094"></a><span class="lineno"> 1094</span> <span class="comment"> *</span></div><div class="line"><a name="l01095"></a><span class="lineno"> 1095</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l01096"></a><span class="lineno"> 1096</span> <span class="comment"> * @param __n The integral order <tt> n >= 0 </tt></span></div><div class="line"><a name="l01097"></a><span class="lineno"> 1097</span> <span class="comment"> * @param __x The real argument <tt> x >= 0 </tt></span></div><div class="line"><a name="l01098"></a><span class="lineno"> 1098</span> <span class="comment"> * @throw std::domain_error if <tt> __x < 0 </tt>.</span></div><div class="line"><a name="l01099"></a><span class="lineno"> 1099</span> <span class="comment"> */</span></div><div class="line"><a name="l01100"></a><span class="lineno"> 1100</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l01101"></a><span class="lineno"> 1101</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l01102"></a><span class="lineno"><a class="line" href="a01497.html#ga478e517ed975bcb256de230e64f0fda5"> 1102</a></span>  <a class="code" href="a01497.html#ga478e517ed975bcb256de230e64f0fda5">sph_bessel</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, _Tp __x)</div><div class="line"><a name="l01103"></a><span class="lineno"> 1103</span>  {</div><div class="line"><a name="l01104"></a><span class="lineno"> 1104</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l01105"></a><span class="lineno"> 1105</span>  <span class="keywordflow">return</span> __detail::__sph_bessel<__type>(__n, __x);</div><div class="line"><a name="l01106"></a><span class="lineno"> 1106</span>  }</div><div class="line"><a name="l01107"></a><span class="lineno"> 1107</span> </div><div class="line"><a name="l01108"></a><span class="lineno"> 1108</span>  <span class="comment">// Spherical associated Legendre functions</span></div><div class="line"><a name="l01109"></a><span class="lineno"> 1109</span> <span class="comment"></span></div><div class="line"><a name="l01110"></a><span class="lineno"> 1110</span> <span class="comment"> /**</span></div><div class="line"><a name="l01111"></a><span class="lineno"> 1111</span> <span class="comment"> * Return the spherical Legendre function of nonnegative integral</span></div><div class="line"><a name="l01112"></a><span class="lineno"> 1112</span> <span class="comment"> * degree @c l and order @c m and float angle @f$ \theta @f$ in radians.</span></div><div class="line"><a name="l01113"></a><span class="lineno"> 1113</span> <span class="comment"> *</span></div><div class="line"><a name="l01114"></a><span class="lineno"> 1114</span> <span class="comment"> * @see sph_legendre for details.</span></div><div class="line"><a name="l01115"></a><span class="lineno"> 1115</span> <span class="comment"> */</span></div><div class="line"><a name="l01116"></a><span class="lineno"> 1116</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l01117"></a><span class="lineno"><a class="line" href="a01497.html#gaae635d28c06a3be2679901b382090852"> 1117</a></span>  <a class="code" href="a01497.html#gaae635d28c06a3be2679901b382090852">sph_legendref</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __l, <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __m, <span class="keywordtype">float</span> __theta)</div><div class="line"><a name="l01118"></a><span class="lineno"> 1118</span>  { <span class="keywordflow">return</span> __detail::__sph_legendre<float>(__l, __m, __theta); }</div><div class="line"><a name="l01119"></a><span class="lineno"> 1119</span> <span class="comment"></span></div><div class="line"><a name="l01120"></a><span class="lineno"> 1120</span> <span class="comment"> /**</span></div><div class="line"><a name="l01121"></a><span class="lineno"> 1121</span> <span class="comment"> * Return the spherical Legendre function of nonnegative integral</span></div><div class="line"><a name="l01122"></a><span class="lineno"> 1122</span> <span class="comment"> * degree @c l and order @c m and <tt>long double</tt> angle @f$ \theta @f$</span></div><div class="line"><a name="l01123"></a><span class="lineno"> 1123</span> <span class="comment"> * in radians.</span></div><div class="line"><a name="l01124"></a><span class="lineno"> 1124</span> <span class="comment"> *</span></div><div class="line"><a name="l01125"></a><span class="lineno"> 1125</span> <span class="comment"> * @see sph_legendre for details.</span></div><div class="line"><a name="l01126"></a><span class="lineno"> 1126</span> <span class="comment"> */</span></div><div class="line"><a name="l01127"></a><span class="lineno"> 1127</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l01128"></a><span class="lineno"><a class="line" href="a01497.html#ga2f6618dea1847f09fd67f3c974c1910d"> 1128</a></span>  <a class="code" href="a01497.html#ga2f6618dea1847f09fd67f3c974c1910d">sph_legendrel</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __l, <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __m, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __theta)</div><div class="line"><a name="l01129"></a><span class="lineno"> 1129</span>  { <span class="keywordflow">return</span> __detail::__sph_legendre<long double>(__l, __m, __theta); }</div><div class="line"><a name="l01130"></a><span class="lineno"> 1130</span> <span class="comment"></span></div><div class="line"><a name="l01131"></a><span class="lineno"> 1131</span> <span class="comment"> /**</span></div><div class="line"><a name="l01132"></a><span class="lineno"> 1132</span> <span class="comment"> * Return the spherical Legendre function of nonnegative integral</span></div><div class="line"><a name="l01133"></a><span class="lineno"> 1133</span> <span class="comment"> * degree @c l and order @c m and real angle @f$ \theta @f$ in radians.</span></div><div class="line"><a name="l01134"></a><span class="lineno"> 1134</span> <span class="comment"> *</span></div><div class="line"><a name="l01135"></a><span class="lineno"> 1135</span> <span class="comment"> * The spherical Legendre function is defined by</span></div><div class="line"><a name="l01136"></a><span class="lineno"> 1136</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l01137"></a><span class="lineno"> 1137</span> <span class="comment"> * Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi}</span></div><div class="line"><a name="l01138"></a><span class="lineno"> 1138</span> <span class="comment"> * \frac{(l-m)!}{(l+m)!}]</span></div><div class="line"><a name="l01139"></a><span class="lineno"> 1139</span> <span class="comment"> * P_l^m(\cos\theta) \exp^{im\phi}</span></div><div class="line"><a name="l01140"></a><span class="lineno"> 1140</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l01141"></a><span class="lineno"> 1141</span> <span class="comment"> *</span></div><div class="line"><a name="l01142"></a><span class="lineno"> 1142</span> <span class="comment"> * @tparam _Tp The floating-point type of the angle @c __theta.</span></div><div class="line"><a name="l01143"></a><span class="lineno"> 1143</span> <span class="comment"> * @param __l The order <tt> __l >= 0 </tt></span></div><div class="line"><a name="l01144"></a><span class="lineno"> 1144</span> <span class="comment"> * @param __m The degree <tt> __m >= 0 </tt> and <tt> __m <= __l </tt></span></div><div class="line"><a name="l01145"></a><span class="lineno"> 1145</span> <span class="comment"> * @param __theta The radian polar angle argument</span></div><div class="line"><a name="l01146"></a><span class="lineno"> 1146</span> <span class="comment"> */</span></div><div class="line"><a name="l01147"></a><span class="lineno"> 1147</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l01148"></a><span class="lineno"> 1148</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l01149"></a><span class="lineno"><a class="line" href="a01497.html#ga573842c12247b87746b548f1945755a8"> 1149</a></span>  <a class="code" href="a01497.html#ga573842c12247b87746b548f1945755a8">sph_legendre</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __l, <span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __m, _Tp __theta)</div><div class="line"><a name="l01150"></a><span class="lineno"> 1150</span>  {</div><div class="line"><a name="l01151"></a><span class="lineno"> 1151</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l01152"></a><span class="lineno"> 1152</span>  <span class="keywordflow">return</span> __detail::__sph_legendre<__type>(__l, __m, __theta);</div><div class="line"><a name="l01153"></a><span class="lineno"> 1153</span>  }</div><div class="line"><a name="l01154"></a><span class="lineno"> 1154</span> </div><div class="line"><a name="l01155"></a><span class="lineno"> 1155</span>  <span class="comment">// Spherical Neumann functions</span></div><div class="line"><a name="l01156"></a><span class="lineno"> 1156</span> <span class="comment"></span></div><div class="line"><a name="l01157"></a><span class="lineno"> 1157</span> <span class="comment"> /**</span></div><div class="line"><a name="l01158"></a><span class="lineno"> 1158</span> <span class="comment"> * Return the spherical Neumann function of integral order @f$ n >= 0 @f$</span></div><div class="line"><a name="l01159"></a><span class="lineno"> 1159</span> <span class="comment"> * and @c float argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l01160"></a><span class="lineno"> 1160</span> <span class="comment"> *</span></div><div class="line"><a name="l01161"></a><span class="lineno"> 1161</span> <span class="comment"> * @see sph_neumann for details.</span></div><div class="line"><a name="l01162"></a><span class="lineno"> 1162</span> <span class="comment"> */</span></div><div class="line"><a name="l01163"></a><span class="lineno"> 1163</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l01164"></a><span class="lineno"><a class="line" href="a01497.html#ga789143122fa99536329bc2d1b1aac2f0"> 1164</a></span>  <a class="code" href="a01497.html#ga789143122fa99536329bc2d1b1aac2f0">sph_neumannf</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l01165"></a><span class="lineno"> 1165</span>  { <span class="keywordflow">return</span> __detail::__sph_neumann<float>(__n, __x); }</div><div class="line"><a name="l01166"></a><span class="lineno"> 1166</span> <span class="comment"></span></div><div class="line"><a name="l01167"></a><span class="lineno"> 1167</span> <span class="comment"> /**</span></div><div class="line"><a name="l01168"></a><span class="lineno"> 1168</span> <span class="comment"> * Return the spherical Neumann function of integral order @f$ n >= 0 @f$</span></div><div class="line"><a name="l01169"></a><span class="lineno"> 1169</span> <span class="comment"> * and <tt>long double</tt> @f$ x >= 0 @f$.</span></div><div class="line"><a name="l01170"></a><span class="lineno"> 1170</span> <span class="comment"> *</span></div><div class="line"><a name="l01171"></a><span class="lineno"> 1171</span> <span class="comment"> * @see sph_neumann for details.</span></div><div class="line"><a name="l01172"></a><span class="lineno"> 1172</span> <span class="comment"> */</span></div><div class="line"><a name="l01173"></a><span class="lineno"> 1173</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l01174"></a><span class="lineno"><a class="line" href="a01497.html#ga3cededa9b6e4601f190c3811e6aabfd6"> 1174</a></span>  <a class="code" href="a01497.html#ga3cededa9b6e4601f190c3811e6aabfd6">sph_neumannl</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l01175"></a><span class="lineno"> 1175</span>  { <span class="keywordflow">return</span> __detail::__sph_neumann<long double>(__n, __x); }</div><div class="line"><a name="l01176"></a><span class="lineno"> 1176</span> <span class="comment"></span></div><div class="line"><a name="l01177"></a><span class="lineno"> 1177</span> <span class="comment"> /**</span></div><div class="line"><a name="l01178"></a><span class="lineno"> 1178</span> <span class="comment"> * Return the spherical Neumann function of integral order @f$ n >= 0 @f$</span></div><div class="line"><a name="l01179"></a><span class="lineno"> 1179</span> <span class="comment"> * and real argument @f$ x >= 0 @f$.</span></div><div class="line"><a name="l01180"></a><span class="lineno"> 1180</span> <span class="comment"> *</span></div><div class="line"><a name="l01181"></a><span class="lineno"> 1181</span> <span class="comment"> * The spherical Neumann function is defined by</span></div><div class="line"><a name="l01182"></a><span class="lineno"> 1182</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l01183"></a><span class="lineno"> 1183</span> <span class="comment"> * n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x)</span></div><div class="line"><a name="l01184"></a><span class="lineno"> 1184</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l01185"></a><span class="lineno"> 1185</span> <span class="comment"> *</span></div><div class="line"><a name="l01186"></a><span class="lineno"> 1186</span> <span class="comment"> * @tparam _Tp The floating-point type of the argument @c __x.</span></div><div class="line"><a name="l01187"></a><span class="lineno"> 1187</span> <span class="comment"> * @param __n The integral order <tt> n >= 0 </tt></span></div><div class="line"><a name="l01188"></a><span class="lineno"> 1188</span> <span class="comment"> * @param __x The real argument <tt> __x >= 0 </tt></span></div><div class="line"><a name="l01189"></a><span class="lineno"> 1189</span> <span class="comment"> * @throw std::domain_error if <tt> __x < 0 </tt>.</span></div><div class="line"><a name="l01190"></a><span class="lineno"> 1190</span> <span class="comment"> */</span></div><div class="line"><a name="l01191"></a><span class="lineno"> 1191</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l01192"></a><span class="lineno"> 1192</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l01193"></a><span class="lineno"><a class="line" href="a01497.html#ga1cf4362a67ab5bae9e6b69c867e85371"> 1193</a></span>  <a class="code" href="a01497.html#ga1cf4362a67ab5bae9e6b69c867e85371">sph_neumann</a>(<span class="keywordtype">unsigned</span> <span class="keywordtype">int</span> __n, _Tp __x)</div><div class="line"><a name="l01194"></a><span class="lineno"> 1194</span>  {</div><div class="line"><a name="l01195"></a><span class="lineno"> 1195</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l01196"></a><span class="lineno"> 1196</span>  <span class="keywordflow">return</span> __detail::__sph_neumann<__type>(__n, __x);</div><div class="line"><a name="l01197"></a><span class="lineno"> 1197</span>  }</div><div class="line"><a name="l01198"></a><span class="lineno"> 1198</span> </div><div class="line"><a name="l01199"></a><span class="lineno"> 1199</span>  <span class="comment">// @} group mathsf</span></div><div class="line"><a name="l01200"></a><span class="lineno"> 1200</span> </div><div class="line"><a name="l01201"></a><span class="lineno"> 1201</span> _GLIBCXX_END_NAMESPACE_VERSION</div><div class="line"><a name="l01202"></a><span class="lineno"> 1202</span> } <span class="comment">// namespace std</span></div><div class="line"><a name="l01203"></a><span class="lineno"> 1203</span> </div><div class="line"><a name="l01204"></a><span class="lineno"> 1204</span> <span class="preprocessor">#ifndef __STRICT_ANSI__</span></div><div class="line"><a name="l01205"></a><span class="lineno"> 1205</span> <span class="keyword">namespace </span><a class="code" href="a01547.html">__gnu_cxx</a> _GLIBCXX_VISIBILITY(default)</div><div class="line"><a name="l01206"></a><span class="lineno"> 1206</span> {</div><div class="line"><a name="l01207"></a><span class="lineno"> 1207</span> _GLIBCXX_BEGIN_NAMESPACE_VERSION</div><div class="line"><a name="l01208"></a><span class="lineno"> 1208</span> </div><div class="line"><a name="l01209"></a><span class="lineno"> 1209</span>  <span class="comment">// Airy functions</span></div><div class="line"><a name="l01210"></a><span class="lineno"> 1210</span> <span class="comment"></span></div><div class="line"><a name="l01211"></a><span class="lineno"> 1211</span> <span class="comment"> /**</span></div><div class="line"><a name="l01212"></a><span class="lineno"> 1212</span> <span class="comment"> * Return the Airy function @f$ Ai(x) @f$ of @c float argument x.</span></div><div class="line"><a name="l01213"></a><span class="lineno"> 1213</span> <span class="comment"> */</span></div><div class="line"><a name="l01214"></a><span class="lineno"> 1214</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l01215"></a><span class="lineno"><a class="line" href="a01547.html#af317ba724c44b3a8271fe341d9870173"> 1215</a></span>  <a class="code" href="a01547.html#af317ba724c44b3a8271fe341d9870173">airy_aif</a>(<span class="keywordtype">float</span> __x)</div><div class="line"><a name="l01216"></a><span class="lineno"> 1216</span>  {</div><div class="line"><a name="l01217"></a><span class="lineno"> 1217</span>  <span class="keywordtype">float</span> __Ai, __Bi, __Aip, __Bip;</div><div class="line"><a name="l01218"></a><span class="lineno"> 1218</span>  std::__detail::__airy<float>(__x, __Ai, __Bi, __Aip, __Bip);</div><div class="line"><a name="l01219"></a><span class="lineno"> 1219</span>  <span class="keywordflow">return</span> __Ai;</div><div class="line"><a name="l01220"></a><span class="lineno"> 1220</span>  }</div><div class="line"><a name="l01221"></a><span class="lineno"> 1221</span> <span class="comment"></span></div><div class="line"><a name="l01222"></a><span class="lineno"> 1222</span> <span class="comment"> /**</span></div><div class="line"><a name="l01223"></a><span class="lineno"> 1223</span> <span class="comment"> * Return the Airy function @f$ Ai(x) @f$ of <tt>long double</tt> argument x.</span></div><div class="line"><a name="l01224"></a><span class="lineno"> 1224</span> <span class="comment"> */</span></div><div class="line"><a name="l01225"></a><span class="lineno"> 1225</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l01226"></a><span class="lineno"><a class="line" href="a01547.html#a800fdb61c672ae1831f4ca4250d657de"> 1226</a></span>  <a class="code" href="a01547.html#a800fdb61c672ae1831f4ca4250d657de">airy_ail</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l01227"></a><span class="lineno"> 1227</span>  {</div><div class="line"><a name="l01228"></a><span class="lineno"> 1228</span>  <span class="keywordtype">long</span> <span class="keywordtype">double</span> __Ai, __Bi, __Aip, __Bip;</div><div class="line"><a name="l01229"></a><span class="lineno"> 1229</span>  std::__detail::__airy<long double>(__x, __Ai, __Bi, __Aip, __Bip);</div><div class="line"><a name="l01230"></a><span class="lineno"> 1230</span>  <span class="keywordflow">return</span> __Ai;</div><div class="line"><a name="l01231"></a><span class="lineno"> 1231</span>  }</div><div class="line"><a name="l01232"></a><span class="lineno"> 1232</span> <span class="comment"></span></div><div class="line"><a name="l01233"></a><span class="lineno"> 1233</span> <span class="comment"> /**</span></div><div class="line"><a name="l01234"></a><span class="lineno"> 1234</span> <span class="comment"> * Return the Airy function @f$ Ai(x) @f$ of real argument x.</span></div><div class="line"><a name="l01235"></a><span class="lineno"> 1235</span> <span class="comment"> */</span></div><div class="line"><a name="l01236"></a><span class="lineno"> 1236</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l01237"></a><span class="lineno"> 1237</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l01238"></a><span class="lineno"><a class="line" href="a01547.html#a3dc92fbf0a20f425585e811e9adb432d"> 1238</a></span>  <a class="code" href="a01547.html#a3dc92fbf0a20f425585e811e9adb432d">airy_ai</a>(_Tp __x)</div><div class="line"><a name="l01239"></a><span class="lineno"> 1239</span>  {</div><div class="line"><a name="l01240"></a><span class="lineno"> 1240</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l01241"></a><span class="lineno"> 1241</span>  __type __Ai, __Bi, __Aip, __Bip;</div><div class="line"><a name="l01242"></a><span class="lineno"> 1242</span>  std::__detail::__airy<__type>(__x, __Ai, __Bi, __Aip, __Bip);</div><div class="line"><a name="l01243"></a><span class="lineno"> 1243</span>  <span class="keywordflow">return</span> __Ai;</div><div class="line"><a name="l01244"></a><span class="lineno"> 1244</span>  }</div><div class="line"><a name="l01245"></a><span class="lineno"> 1245</span> <span class="comment"></span></div><div class="line"><a name="l01246"></a><span class="lineno"> 1246</span> <span class="comment"> /**</span></div><div class="line"><a name="l01247"></a><span class="lineno"> 1247</span> <span class="comment"> * Return the Airy function @f$ Bi(x) @f$ of @c float argument x.</span></div><div class="line"><a name="l01248"></a><span class="lineno"> 1248</span> <span class="comment"> */</span></div><div class="line"><a name="l01249"></a><span class="lineno"> 1249</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l01250"></a><span class="lineno"><a class="line" href="a01547.html#a2ade465827bdba7370abbcce78e54912"> 1250</a></span>  <a class="code" href="a01547.html#a2ade465827bdba7370abbcce78e54912">airy_bif</a>(<span class="keywordtype">float</span> __x)</div><div class="line"><a name="l01251"></a><span class="lineno"> 1251</span>  {</div><div class="line"><a name="l01252"></a><span class="lineno"> 1252</span>  <span class="keywordtype">float</span> __Ai, __Bi, __Aip, __Bip;</div><div class="line"><a name="l01253"></a><span class="lineno"> 1253</span>  std::__detail::__airy<float>(__x, __Ai, __Bi, __Aip, __Bip);</div><div class="line"><a name="l01254"></a><span class="lineno"> 1254</span>  <span class="keywordflow">return</span> __Bi;</div><div class="line"><a name="l01255"></a><span class="lineno"> 1255</span>  }</div><div class="line"><a name="l01256"></a><span class="lineno"> 1256</span> <span class="comment"></span></div><div class="line"><a name="l01257"></a><span class="lineno"> 1257</span> <span class="comment"> /**</span></div><div class="line"><a name="l01258"></a><span class="lineno"> 1258</span> <span class="comment"> * Return the Airy function @f$ Bi(x) @f$ of <tt>long double</tt> argument x.</span></div><div class="line"><a name="l01259"></a><span class="lineno"> 1259</span> <span class="comment"> */</span></div><div class="line"><a name="l01260"></a><span class="lineno"> 1260</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l01261"></a><span class="lineno"><a class="line" href="a01547.html#a59240b3f40177e5187f3f194f624f0f8"> 1261</a></span>  <a class="code" href="a01547.html#a59240b3f40177e5187f3f194f624f0f8">airy_bil</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l01262"></a><span class="lineno"> 1262</span>  {</div><div class="line"><a name="l01263"></a><span class="lineno"> 1263</span>  <span class="keywordtype">long</span> <span class="keywordtype">double</span> __Ai, __Bi, __Aip, __Bip;</div><div class="line"><a name="l01264"></a><span class="lineno"> 1264</span>  std::__detail::__airy<long double>(__x, __Ai, __Bi, __Aip, __Bip);</div><div class="line"><a name="l01265"></a><span class="lineno"> 1265</span>  <span class="keywordflow">return</span> __Bi;</div><div class="line"><a name="l01266"></a><span class="lineno"> 1266</span>  }</div><div class="line"><a name="l01267"></a><span class="lineno"> 1267</span> <span class="comment"></span></div><div class="line"><a name="l01268"></a><span class="lineno"> 1268</span> <span class="comment"> /**</span></div><div class="line"><a name="l01269"></a><span class="lineno"> 1269</span> <span class="comment"> * Return the Airy function @f$ Bi(x) @f$ of real argument x.</span></div><div class="line"><a name="l01270"></a><span class="lineno"> 1270</span> <span class="comment"> */</span></div><div class="line"><a name="l01271"></a><span class="lineno"> 1271</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tp></div><div class="line"><a name="l01272"></a><span class="lineno"> 1272</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type</div><div class="line"><a name="l01273"></a><span class="lineno"><a class="line" href="a01547.html#accafc84b7c86a0c99b82f88eb4b1a43e"> 1273</a></span>  <a class="code" href="a01547.html#accafc84b7c86a0c99b82f88eb4b1a43e">airy_bi</a>(_Tp __x)</div><div class="line"><a name="l01274"></a><span class="lineno"> 1274</span>  {</div><div class="line"><a name="l01275"></a><span class="lineno"> 1275</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote<_Tp>::__type __type;</div><div class="line"><a name="l01276"></a><span class="lineno"> 1276</span>  __type __Ai, __Bi, __Aip, __Bip;</div><div class="line"><a name="l01277"></a><span class="lineno"> 1277</span>  std::__detail::__airy<__type>(__x, __Ai, __Bi, __Aip, __Bip);</div><div class="line"><a name="l01278"></a><span class="lineno"> 1278</span>  <span class="keywordflow">return</span> __Bi;</div><div class="line"><a name="l01279"></a><span class="lineno"> 1279</span>  }</div><div class="line"><a name="l01280"></a><span class="lineno"> 1280</span> </div><div class="line"><a name="l01281"></a><span class="lineno"> 1281</span>  <span class="comment">// Confluent hypergeometric functions</span></div><div class="line"><a name="l01282"></a><span class="lineno"> 1282</span> <span class="comment"></span></div><div class="line"><a name="l01283"></a><span class="lineno"> 1283</span> <span class="comment"> /**</span></div><div class="line"><a name="l01284"></a><span class="lineno"> 1284</span> <span class="comment"> * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$</span></div><div class="line"><a name="l01285"></a><span class="lineno"> 1285</span> <span class="comment"> * of @c float numeratorial parameter @c a, denominatorial parameter @c c,</span></div><div class="line"><a name="l01286"></a><span class="lineno"> 1286</span> <span class="comment"> * and argument @c x.</span></div><div class="line"><a name="l01287"></a><span class="lineno"> 1287</span> <span class="comment"> *</span></div><div class="line"><a name="l01288"></a><span class="lineno"> 1288</span> <span class="comment"> * @see conf_hyperg for details.</span></div><div class="line"><a name="l01289"></a><span class="lineno"> 1289</span> <span class="comment"> */</span></div><div class="line"><a name="l01290"></a><span class="lineno"> 1290</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l01291"></a><span class="lineno"><a class="line" href="a01547.html#abd18e600aa78c3f2b2f835039506c810"> 1291</a></span>  <a class="code" href="a01547.html#abd18e600aa78c3f2b2f835039506c810">conf_hypergf</a>(<span class="keywordtype">float</span> __a, <span class="keywordtype">float</span> __c, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l01292"></a><span class="lineno"> 1292</span>  { <span class="keywordflow">return</span> std::__detail::__conf_hyperg<float>(__a, __c, __x); }</div><div class="line"><a name="l01293"></a><span class="lineno"> 1293</span> <span class="comment"></span></div><div class="line"><a name="l01294"></a><span class="lineno"> 1294</span> <span class="comment"> /**</span></div><div class="line"><a name="l01295"></a><span class="lineno"> 1295</span> <span class="comment"> * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$</span></div><div class="line"><a name="l01296"></a><span class="lineno"> 1296</span> <span class="comment"> * of <tt>long double</tt> numeratorial parameter @c a,</span></div><div class="line"><a name="l01297"></a><span class="lineno"> 1297</span> <span class="comment"> * denominatorial parameter @c c, and argument @c x.</span></div><div class="line"><a name="l01298"></a><span class="lineno"> 1298</span> <span class="comment"> *</span></div><div class="line"><a name="l01299"></a><span class="lineno"> 1299</span> <span class="comment"> * @see conf_hyperg for details.</span></div><div class="line"><a name="l01300"></a><span class="lineno"> 1300</span> <span class="comment"> */</span></div><div class="line"><a name="l01301"></a><span class="lineno"> 1301</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l01302"></a><span class="lineno"><a class="line" href="a01547.html#a0a9853f30d8fa515a12cd45a92da832e"> 1302</a></span>  <a class="code" href="a01547.html#a0a9853f30d8fa515a12cd45a92da832e">conf_hypergl</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __a, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __c, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l01303"></a><span class="lineno"> 1303</span>  { <span class="keywordflow">return</span> std::__detail::__conf_hyperg<long double>(__a, __c, __x); }</div><div class="line"><a name="l01304"></a><span class="lineno"> 1304</span> <span class="comment"></span></div><div class="line"><a name="l01305"></a><span class="lineno"> 1305</span> <span class="comment"> /**</span></div><div class="line"><a name="l01306"></a><span class="lineno"> 1306</span> <span class="comment"> * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$</span></div><div class="line"><a name="l01307"></a><span class="lineno"> 1307</span> <span class="comment"> * of real numeratorial parameter @c a, denominatorial parameter @c c,</span></div><div class="line"><a name="l01308"></a><span class="lineno"> 1308</span> <span class="comment"> * and argument @c x.</span></div><div class="line"><a name="l01309"></a><span class="lineno"> 1309</span> <span class="comment"> *</span></div><div class="line"><a name="l01310"></a><span class="lineno"> 1310</span> <span class="comment"> * The confluent hypergeometric function is defined by</span></div><div class="line"><a name="l01311"></a><span class="lineno"> 1311</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l01312"></a><span class="lineno"> 1312</span> <span class="comment"> * {}_1F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n x^n}{(c)_n n!}</span></div><div class="line"><a name="l01313"></a><span class="lineno"> 1313</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l01314"></a><span class="lineno"> 1314</span> <span class="comment"> * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$,</span></div><div class="line"><a name="l01315"></a><span class="lineno"> 1315</span> <span class="comment"> * @f$ (x)_0 = 1 @f$</span></div><div class="line"><a name="l01316"></a><span class="lineno"> 1316</span> <span class="comment"> *</span></div><div class="line"><a name="l01317"></a><span class="lineno"> 1317</span> <span class="comment"> * @param __a The numeratorial parameter</span></div><div class="line"><a name="l01318"></a><span class="lineno"> 1318</span> <span class="comment"> * @param __c The denominatorial parameter</span></div><div class="line"><a name="l01319"></a><span class="lineno"> 1319</span> <span class="comment"> * @param __x The argument</span></div><div class="line"><a name="l01320"></a><span class="lineno"> 1320</span> <span class="comment"> */</span></div><div class="line"><a name="l01321"></a><span class="lineno"> 1321</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tpa, <span class="keyword">typename</span> _Tpc, <span class="keyword">typename</span> _Tp></div><div class="line"><a name="l01322"></a><span class="lineno"> 1322</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type</div><div class="line"><a name="l01323"></a><span class="lineno"><a class="line" href="a01547.html#a2e17ccbbc4cbb99c987e875531d4a3de"> 1323</a></span>  <a class="code" href="a01547.html#a2e17ccbbc4cbb99c987e875531d4a3de">conf_hyperg</a>(_Tpa __a, _Tpc __c, _Tp __x)</div><div class="line"><a name="l01324"></a><span class="lineno"> 1324</span>  {</div><div class="line"><a name="l01325"></a><span class="lineno"> 1325</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type __type;</div><div class="line"><a name="l01326"></a><span class="lineno"> 1326</span>  <span class="keywordflow">return</span> std::__detail::__conf_hyperg<__type>(__a, __c, __x);</div><div class="line"><a name="l01327"></a><span class="lineno"> 1327</span>  }</div><div class="line"><a name="l01328"></a><span class="lineno"> 1328</span> </div><div class="line"><a name="l01329"></a><span class="lineno"> 1329</span>  <span class="comment">// Hypergeometric functions</span></div><div class="line"><a name="l01330"></a><span class="lineno"> 1330</span> <span class="comment"></span></div><div class="line"><a name="l01331"></a><span class="lineno"> 1331</span> <span class="comment"> /**</span></div><div class="line"><a name="l01332"></a><span class="lineno"> 1332</span> <span class="comment"> * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$</span></div><div class="line"><a name="l01333"></a><span class="lineno"> 1333</span> <span class="comment"> * of @ float numeratorial parameters @c a and @c b,</span></div><div class="line"><a name="l01334"></a><span class="lineno"> 1334</span> <span class="comment"> * denominatorial parameter @c c, and argument @c x.</span></div><div class="line"><a name="l01335"></a><span class="lineno"> 1335</span> <span class="comment"> *</span></div><div class="line"><a name="l01336"></a><span class="lineno"> 1336</span> <span class="comment"> * @see hyperg for details.</span></div><div class="line"><a name="l01337"></a><span class="lineno"> 1337</span> <span class="comment"> */</span></div><div class="line"><a name="l01338"></a><span class="lineno"> 1338</span>  <span class="keyword">inline</span> <span class="keywordtype">float</span></div><div class="line"><a name="l01339"></a><span class="lineno"><a class="line" href="a01547.html#ac4c81e4ea9cef149fe40291ca10d7e15"> 1339</a></span>  <a class="code" href="a01547.html#ac4c81e4ea9cef149fe40291ca10d7e15">hypergf</a>(<span class="keywordtype">float</span> __a, <span class="keywordtype">float</span> __b, <span class="keywordtype">float</span> __c, <span class="keywordtype">float</span> __x)</div><div class="line"><a name="l01340"></a><span class="lineno"> 1340</span>  { <span class="keywordflow">return</span> std::__detail::__hyperg<float>(__a, __b, __c, __x); }</div><div class="line"><a name="l01341"></a><span class="lineno"> 1341</span> <span class="comment"></span></div><div class="line"><a name="l01342"></a><span class="lineno"> 1342</span> <span class="comment"> /**</span></div><div class="line"><a name="l01343"></a><span class="lineno"> 1343</span> <span class="comment"> * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$</span></div><div class="line"><a name="l01344"></a><span class="lineno"> 1344</span> <span class="comment"> * of <tt>long double</tt> numeratorial parameters @c a and @c b,</span></div><div class="line"><a name="l01345"></a><span class="lineno"> 1345</span> <span class="comment"> * denominatorial parameter @c c, and argument @c x.</span></div><div class="line"><a name="l01346"></a><span class="lineno"> 1346</span> <span class="comment"> *</span></div><div class="line"><a name="l01347"></a><span class="lineno"> 1347</span> <span class="comment"> * @see hyperg for details.</span></div><div class="line"><a name="l01348"></a><span class="lineno"> 1348</span> <span class="comment"> */</span></div><div class="line"><a name="l01349"></a><span class="lineno"> 1349</span>  <span class="keyword">inline</span> <span class="keywordtype">long</span> <span class="keywordtype">double</span></div><div class="line"><a name="l01350"></a><span class="lineno"><a class="line" href="a01547.html#a9961967087216e97f76283f29e1be152"> 1350</a></span>  <a class="code" href="a01547.html#a9961967087216e97f76283f29e1be152">hypergl</a>(<span class="keywordtype">long</span> <span class="keywordtype">double</span> __a, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __b, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __c, <span class="keywordtype">long</span> <span class="keywordtype">double</span> __x)</div><div class="line"><a name="l01351"></a><span class="lineno"> 1351</span>  { <span class="keywordflow">return</span> std::__detail::__hyperg<long double>(__a, __b, __c, __x); }</div><div class="line"><a name="l01352"></a><span class="lineno"> 1352</span> <span class="comment"></span></div><div class="line"><a name="l01353"></a><span class="lineno"> 1353</span> <span class="comment"> /**</span></div><div class="line"><a name="l01354"></a><span class="lineno"> 1354</span> <span class="comment"> * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$</span></div><div class="line"><a name="l01355"></a><span class="lineno"> 1355</span> <span class="comment"> * of real numeratorial parameters @c a and @c b,</span></div><div class="line"><a name="l01356"></a><span class="lineno"> 1356</span> <span class="comment"> * denominatorial parameter @c c, and argument @c x.</span></div><div class="line"><a name="l01357"></a><span class="lineno"> 1357</span> <span class="comment"> *</span></div><div class="line"><a name="l01358"></a><span class="lineno"> 1358</span> <span class="comment"> * The hypergeometric function is defined by</span></div><div class="line"><a name="l01359"></a><span class="lineno"> 1359</span> <span class="comment"> * @f[</span></div><div class="line"><a name="l01360"></a><span class="lineno"> 1360</span> <span class="comment"> * {}_2F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n x^n}{(c)_n n!}</span></div><div class="line"><a name="l01361"></a><span class="lineno"> 1361</span> <span class="comment"> * @f]</span></div><div class="line"><a name="l01362"></a><span class="lineno"> 1362</span> <span class="comment"> * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$,</span></div><div class="line"><a name="l01363"></a><span class="lineno"> 1363</span> <span class="comment"> * @f$ (x)_0 = 1 @f$</span></div><div class="line"><a name="l01364"></a><span class="lineno"> 1364</span> <span class="comment"> *</span></div><div class="line"><a name="l01365"></a><span class="lineno"> 1365</span> <span class="comment"> * @param __a The first numeratorial parameter</span></div><div class="line"><a name="l01366"></a><span class="lineno"> 1366</span> <span class="comment"> * @param __b The second numeratorial parameter</span></div><div class="line"><a name="l01367"></a><span class="lineno"> 1367</span> <span class="comment"> * @param __c The denominatorial parameter</span></div><div class="line"><a name="l01368"></a><span class="lineno"> 1368</span> <span class="comment"> * @param __x The argument</span></div><div class="line"><a name="l01369"></a><span class="lineno"> 1369</span> <span class="comment"> */</span></div><div class="line"><a name="l01370"></a><span class="lineno"> 1370</span>  <span class="keyword">template</span><<span class="keyword">typename</span> _Tpa, <span class="keyword">typename</span> _Tpb, <span class="keyword">typename</span> _Tpc, <span class="keyword">typename</span> _Tp></div><div class="line"><a name="l01371"></a><span class="lineno"> 1371</span>  <span class="keyword">inline</span> <span class="keyword">typename</span> __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp>::__type</div><div class="line"><a name="l01372"></a><span class="lineno"><a class="line" href="a01547.html#af52cf49736c63b0bb000be98b53c221f"> 1372</a></span>  <a class="code" href="a01547.html#af52cf49736c63b0bb000be98b53c221f">hyperg</a>(_Tpa __a, _Tpb __b, _Tpc __c, _Tp __x)</div><div class="line"><a name="l01373"></a><span class="lineno"> 1373</span>  {</div><div class="line"><a name="l01374"></a><span class="lineno"> 1374</span>  <span class="keyword">typedef</span> <span class="keyword">typename</span> __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp></div><div class="line"><a name="l01375"></a><span class="lineno"> 1375</span>  ::__type __type;</div><div class="line"><a name="l01376"></a><span class="lineno"> 1376</span>  <span class="keywordflow">return</span> std::__detail::__hyperg<__type>(__a, __b, __c, __x);</div><div class="line"><a name="l01377"></a><span class="lineno"> 1377</span>  }</div><div class="line"><a name="l01378"></a><span class="lineno"> 1378</span> </div><div class="line"><a name="l01379"></a><span class="lineno"> 1379</span> _GLIBCXX_END_NAMESPACE_VERSION</div><div class="line"><a name="l01380"></a><span class="lineno"> 1380</span> } <span class="comment">// namespace __gnu_cxx</span></div><div class="line"><a name="l01381"></a><span class="lineno"> 1381</span> <span class="preprocessor">#endif // __STRICT_ANSI__</span></div><div class="line"><a name="l01382"></a><span class="lineno"> 1382</span> </div><div class="line"><a name="l01383"></a><span class="lineno"> 1383</span> <span class="preprocessor">#pragma GCC visibility pop</span></div><div class="line"><a name="l01384"></a><span class="lineno"> 1384</span> </div><div class="line"><a name="l01385"></a><span class="lineno"> 1385</span> <span class="preprocessor">#endif // _GLIBCXX_BITS_SPECFUN_H</span></div><div class="ttc" id="a01497_html_gad6dd71db2b3f90d24ff49bf8cf37bc37"><div class="ttname"><a href="a01497.html#gad6dd71db2b3f90d24ff49bf8cf37bc37">std::ellint_2</a></div><div class="ttdeci">__gnu_cxx::__promote_2< _Tp, _Tpp >::__type ellint_2(_Tp __k, _Tpp __phi)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00777">specfun.h:777</a></div></div>
<div class="ttc" id="a01497_html_ga377bb7e038c464a27dfe0573fd2d7b33"><div class="ttname"><a href="a01497.html#ga377bb7e038c464a27dfe0573fd2d7b33">std::assoc_laguerre</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00252">specfun.h:252</a></div></div>
<div class="ttc" id="a01497_html_ga1329130b32328d0666e290ee5931fa4f"><div class="ttname"><a href="a01497.html#ga1329130b32328d0666e290ee5931fa4f">std::expintl</a></div><div class="ttdeci">long double expintl(long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00854">specfun.h:854</a></div></div>
<div class="ttc" id="a01497_html_gaf92063315061a56d3e2c4053156d968e"><div class="ttname"><a href="a01497.html#gaf92063315061a56d3e2c4053156d968e">std::riemann_zetaf</a></div><div class="ttdeci">float riemann_zetaf(float __s)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01022">specfun.h:1022</a></div></div>
<div class="ttc" id="a01497_html_ga355349f79119c1fd1e2a9351cec57f0f"><div class="ttname"><a href="a01497.html#ga355349f79119c1fd1e2a9351cec57f0f">std::assoc_legendre</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type assoc_legendre(unsigned int __l, unsigned int __m, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00298">specfun.h:298</a></div></div>
<div class="ttc" id="a01497_html_ga3ced07ddd24bf4af56e2712d148e7f57"><div class="ttname"><a href="a01497.html#ga3ced07ddd24bf4af56e2712d148e7f57">std::assoc_legendref</a></div><div class="ttdeci">float assoc_legendref(unsigned int __l, unsigned int __m, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00267">specfun.h:267</a></div></div>
<div class="ttc" id="a01497_html_gab7962629216d03efb8ecaa3f70c6878f"><div class="ttname"><a href="a01497.html#gab7962629216d03efb8ecaa3f70c6878f">std::cyl_bessel_il</a></div><div class="ttdeci">long double cyl_bessel_il(long double __nu, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00514">specfun.h:514</a></div></div>
<div class="ttc" id="a01497_html_ga88ba17f5d050a6973ca4db1bf6e90590"><div class="ttname"><a href="a01497.html#ga88ba17f5d050a6973ca4db1bf6e90590">std::expint</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type expint(_Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00870">specfun.h:870</a></div></div>
<div class="ttc" id="a01547_html_accafc84b7c86a0c99b82f88eb4b1a43e"><div class="ttname"><a href="a01547.html#accafc84b7c86a0c99b82f88eb4b1a43e">__gnu_cxx::airy_bi</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type airy_bi(_Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01273">specfun.h:1273</a></div></div>
<div class="ttc" id="a01497_html_gaa8c0e5864df8769021a7f3e21a30c5d2"><div class="ttname"><a href="a01497.html#gaa8c0e5864df8769021a7f3e21a30c5d2">std::ellint_3l</a></div><div class="ttdeci">long double ellint_3l(long double __k, long double __nu, long double __phi)</div><div class="ttdoc">Return the incomplete elliptic integral of the third kind .</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00802">specfun.h:802</a></div></div>
<div class="ttc" id="a00656_html"><div class="ttname"><a href="a00656.html">c++config.h</a></div></div>
<div class="ttc" id="a01547_html_abd18e600aa78c3f2b2f835039506c810"><div class="ttname"><a href="a01547.html#abd18e600aa78c3f2b2f835039506c810">__gnu_cxx::conf_hypergf</a></div><div class="ttdeci">float conf_hypergf(float __a, float __c, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01291">specfun.h:1291</a></div></div>
<div class="ttc" id="a01497_html_ga47e21a13b6d68d0d7f057699bd3b3ce0"><div class="ttname"><a href="a01497.html#ga47e21a13b6d68d0d7f057699bd3b3ce0">std::cyl_bessel_j</a></div><div class="ttdeci">__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type cyl_bessel_j(_Tpnu __nu, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00581">specfun.h:581</a></div></div>
<div class="ttc" id="a01497_html_gaed94e3c664c99f5204da75be75aeac21"><div class="ttname"><a href="a01497.html#gaed94e3c664c99f5204da75be75aeac21">std::legendref</a></div><div class="ttdeci">float legendref(unsigned int __l, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00977">specfun.h:977</a></div></div>
<div class="ttc" id="a01497_html_ga1b39bc22e3cc4860d08eb54099460391"><div class="ttname"><a href="a01497.html#ga1b39bc22e3cc4860d08eb54099460391">std::legendrel</a></div><div class="ttdeci">long double legendrel(unsigned int __l, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00987">specfun.h:987</a></div></div>
<div class="ttc" id="a00530_html"><div class="ttname"><a href="a00530.html">stl_algobase.h</a></div></div>
<div class="ttc" id="a01547_html_a2e17ccbbc4cbb99c987e875531d4a3de"><div class="ttname"><a href="a01547.html#a2e17ccbbc4cbb99c987e875531d4a3de">__gnu_cxx::conf_hyperg</a></div><div class="ttdeci">__gnu_cxx::__promote_3< _Tpa, _Tpc, _Tp >::__type conf_hyperg(_Tpa __a, _Tpc __c, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01323">specfun.h:1323</a></div></div>
<div class="ttc" id="a01547_html_a800fdb61c672ae1831f4ca4250d657de"><div class="ttname"><a href="a01547.html#a800fdb61c672ae1831f4ca4250d657de">__gnu_cxx::airy_ail</a></div><div class="ttdeci">long double airy_ail(long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01226">specfun.h:1226</a></div></div>
<div class="ttc" id="a01497_html_ga5c791332d374a809d8ca16c69a1a30f5"><div class="ttname"><a href="a01497.html#ga5c791332d374a809d8ca16c69a1a30f5">std::ellint_2l</a></div><div class="ttdeci">long double ellint_2l(long double __k, long double __phi)</div><div class="ttdoc">Return the incomplete elliptic integral of the second kind .</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00754">specfun.h:754</a></div></div>
<div class="ttc" id="a01497_html_gac559500c604c43ea943d593c9ad9d289"><div class="ttname"><a href="a01497.html#gac559500c604c43ea943d593c9ad9d289">std::comp_ellint_1</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type comp_ellint_1(_Tp __k)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00391">specfun.h:391</a></div></div>
<div class="ttc" id="a01497_html_ga20832e3a67d25cc3d415cafc88019ac3"><div class="ttname"><a href="a01497.html#ga20832e3a67d25cc3d415cafc88019ac3">std::ellint_3</a></div><div class="ttdeci">__gnu_cxx::__promote_3< _Tp, _Tpn, _Tpp >::__type ellint_3(_Tp __k, _Tpn __nu, _Tpp __phi)</div><div class="ttdoc">Return the incomplete elliptic integral of the third kind .</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00830">specfun.h:830</a></div></div>
<div class="ttc" id="a01497_html_ga3cededa9b6e4601f190c3811e6aabfd6"><div class="ttname"><a href="a01497.html#ga3cededa9b6e4601f190c3811e6aabfd6">std::sph_neumannl</a></div><div class="ttdeci">long double sph_neumannl(unsigned int __n, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01174">specfun.h:1174</a></div></div>
<div class="ttc" id="a01497_html_ga308d23d70f4b5e848eb7a4173628ef3b"><div class="ttname"><a href="a01497.html#ga308d23d70f4b5e848eb7a4173628ef3b">std::ellint_1f</a></div><div class="ttdeci">float ellint_1f(float __k, float __phi)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00696">specfun.h:696</a></div></div>
<div class="ttc" id="a01497_html_ga594a730163c6228c75b152462700062b"><div class="ttname"><a href="a01497.html#ga594a730163c6228c75b152462700062b">std::ellint_2f</a></div><div class="ttdeci">float ellint_2f(float __k, float __phi)</div><div class="ttdoc">Return the incomplete elliptic integral of the second kind for float argument.</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00744">specfun.h:744</a></div></div>
<div class="ttc" id="a01497_html_ga789143122fa99536329bc2d1b1aac2f0"><div class="ttname"><a href="a01497.html#ga789143122fa99536329bc2d1b1aac2f0">std::sph_neumannf</a></div><div class="ttdeci">float sph_neumannf(unsigned int __n, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01164">specfun.h:1164</a></div></div>
<div class="ttc" id="a01497_html_ga94dae7444bb349e33057a92932db8abe"><div class="ttname"><a href="a01497.html#ga94dae7444bb349e33057a92932db8abe">std::hermitef</a></div><div class="ttdeci">float hermitef(unsigned int __n, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00885">specfun.h:885</a></div></div>
<div class="ttc" id="a01497_html_ga97632b8bf77c323b2369e8d327965bdf"><div class="ttname"><a href="a01497.html#ga97632b8bf77c323b2369e8d327965bdf">std::hermite</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type hermite(unsigned int __n, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00918">specfun.h:918</a></div></div>
<div class="ttc" id="a00167_html"><div class="ttname"><a href="a00167.html">type_traits</a></div></div>
<div class="ttc" id="a01497_html_ga8caca1cef099f41a88111209c36ce06c"><div class="ttname"><a href="a01497.html#ga8caca1cef099f41a88111209c36ce06c">std::betal</a></div><div class="ttdeci">long double betal(long double __a, long double __b)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00322">specfun.h:322</a></div></div>
<div class="ttc" id="a01497_html_ga11d72b1af81ce9da3c878a25087ee927"><div class="ttname"><a href="a01497.html#ga11d72b1af81ce9da3c878a25087ee927">std::sph_bessell</a></div><div class="ttdeci">long double sph_bessell(unsigned int __n, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01083">specfun.h:1083</a></div></div>
<div class="ttc" id="a01547_html"><div class="ttname"><a href="a01547.html">__gnu_cxx</a></div><div class="ttdoc">GNU extensions for public use.</div></div>
<div class="ttc" id="a01497_html_ga76dcd3884620955680112aca0d327ada"><div class="ttname"><a href="a01497.html#ga76dcd3884620955680112aca0d327ada">std::cyl_bessel_k</a></div><div class="ttdeci">__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type cyl_bessel_k(_Tpnu __nu, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00633">specfun.h:633</a></div></div>
<div class="ttc" id="a01497_html_ga5842816f6eed2594e0a327df4e4a2a47"><div class="ttname"><a href="a01497.html#ga5842816f6eed2594e0a327df4e4a2a47">std::expintf</a></div><div class="ttdeci">float expintf(float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00844">specfun.h:844</a></div></div>
<div class="ttc" id="a01497_html_ga12dc61ee4c09172151cf092ed387e203"><div class="ttname"><a href="a01497.html#ga12dc61ee4c09172151cf092ed387e203">std::betaf</a></div><div class="ttdeci">float betaf(float __a, float __b)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00312">specfun.h:312</a></div></div>
<div class="ttc" id="a01497_html_ga21f8e312ee3e65286f86bf141b0f32e0"><div class="ttname"><a href="a01497.html#ga21f8e312ee3e65286f86bf141b0f32e0">std::hermitel</a></div><div class="ttdeci">long double hermitel(unsigned int __n, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00895">specfun.h:895</a></div></div>
<div class="ttc" id="a01497_html_ga15731a7bccd6351d28353e3c4c2a2d23"><div class="ttname"><a href="a01497.html#ga15731a7bccd6351d28353e3c4c2a2d23">std::cyl_bessel_jf</a></div><div class="ttdeci">float cyl_bessel_jf(float __nu, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00550">specfun.h:550</a></div></div>
<div class="ttc" id="a01497_html_ga47b647ec386c8d4b18a030c97842df18"><div class="ttname"><a href="a01497.html#ga47b647ec386c8d4b18a030c97842df18">std::comp_ellint_2l</a></div><div class="ttdeci">long double comp_ellint_2l(long double __k)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00416">specfun.h:416</a></div></div>
<div class="ttc" id="a01497_html_ga55977b425a539146f060dec1c8003344"><div class="ttname"><a href="a01497.html#ga55977b425a539146f060dec1c8003344">std::assoc_legendrel</a></div><div class="ttdeci">long double assoc_legendrel(unsigned int __l, unsigned int __m, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00276">specfun.h:276</a></div></div>
<div class="ttc" id="a01541_html"><div class="ttname"><a href="a01541.html">std</a></div><div class="ttdoc">ISO C++ entities toplevel namespace is std.</div></div>
<div class="ttc" id="a00095_html"><div class="ttname"><a href="a00095.html">limits</a></div></div>
<div class="ttc" id="a01497_html_gaf83d98f350a1cfcebee6a1f723cf90d2"><div class="ttname"><a href="a01497.html#gaf83d98f350a1cfcebee6a1f723cf90d2">std::assoc_laguerref</a></div><div class="ttdeci">float assoc_laguerref(unsigned int __n, unsigned int __m, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00206">specfun.h:206</a></div></div>
<div class="ttc" id="a01497_html_ga76834d3112f777703330892303267a39"><div class="ttname"><a href="a01497.html#ga76834d3112f777703330892303267a39">std::comp_ellint_3f</a></div><div class="ttdeci">float comp_ellint_3f(float __k, float __nu)</div><div class="ttdoc">Return the complete elliptic integral of the third kind for float modulus k.</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00453">specfun.h:453</a></div></div>
<div class="ttc" id="a01497_html_ga7fb5be999a8125cf7e55e630eb8444a1"><div class="ttname"><a href="a01497.html#ga7fb5be999a8125cf7e55e630eb8444a1">std::comp_ellint_1f</a></div><div class="ttdeci">float comp_ellint_1f(float __k)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00358">specfun.h:358</a></div></div>
<div class="ttc" id="a01547_html_a2ade465827bdba7370abbcce78e54912"><div class="ttname"><a href="a01547.html#a2ade465827bdba7370abbcce78e54912">__gnu_cxx::airy_bif</a></div><div class="ttdeci">float airy_bif(float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01250">specfun.h:1250</a></div></div>
<div class="ttc" id="a01547_html_a3dc92fbf0a20f425585e811e9adb432d"><div class="ttname"><a href="a01547.html#a3dc92fbf0a20f425585e811e9adb432d">__gnu_cxx::airy_ai</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type airy_ai(_Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01238">specfun.h:1238</a></div></div>
<div class="ttc" id="a01497_html_ga604c13e8f2bb7cd3c7c91d8b19d6b13a"><div class="ttname"><a href="a01497.html#ga604c13e8f2bb7cd3c7c91d8b19d6b13a">std::cyl_neumannf</a></div><div class="ttdeci">float cyl_neumannf(float __nu, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00648">specfun.h:648</a></div></div>
<div class="ttc" id="a01497_html_ga478e517ed975bcb256de230e64f0fda5"><div class="ttname"><a href="a01497.html#ga478e517ed975bcb256de230e64f0fda5">std::sph_bessel</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type sph_bessel(unsigned int __n, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01102">specfun.h:1102</a></div></div>
<div class="ttc" id="a01547_html_ac4c81e4ea9cef149fe40291ca10d7e15"><div class="ttname"><a href="a01547.html#ac4c81e4ea9cef149fe40291ca10d7e15">__gnu_cxx::hypergf</a></div><div class="ttdeci">float hypergf(float __a, float __b, float __c, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01339">specfun.h:1339</a></div></div>
<div class="ttc" id="a01497_html_ga534e36e1dcefad8daec98920db16eec4"><div class="ttname"><a href="a01497.html#ga534e36e1dcefad8daec98920db16eec4">std::sph_besself</a></div><div class="ttdeci">float sph_besself(unsigned int __n, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01073">specfun.h:1073</a></div></div>
<div class="ttc" id="a01497_html_ga573842c12247b87746b548f1945755a8"><div class="ttname"><a href="a01497.html#ga573842c12247b87746b548f1945755a8">std::sph_legendre</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01149">specfun.h:1149</a></div></div>
<div class="ttc" id="a01497_html_ga2f6618dea1847f09fd67f3c974c1910d"><div class="ttname"><a href="a01497.html#ga2f6618dea1847f09fd67f3c974c1910d">std::sph_legendrel</a></div><div class="ttdeci">long double sph_legendrel(unsigned int __l, unsigned int __m, long double __theta)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01128">specfun.h:1128</a></div></div>
<div class="ttc" id="a01497_html_ga21700f2f125c42b1f1da1f9c7eea1135"><div class="ttname"><a href="a01497.html#ga21700f2f125c42b1f1da1f9c7eea1135">std::comp_ellint_2f</a></div><div class="ttdeci">float comp_ellint_2f(float __k)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00406">specfun.h:406</a></div></div>
<div class="ttc" id="a01547_html_a0a9853f30d8fa515a12cd45a92da832e"><div class="ttname"><a href="a01547.html#a0a9853f30d8fa515a12cd45a92da832e">__gnu_cxx::conf_hypergl</a></div><div class="ttdeci">long double conf_hypergl(long double __a, long double __c, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01302">specfun.h:1302</a></div></div>
<div class="ttc" id="a01547_html_a9961967087216e97f76283f29e1be152"><div class="ttname"><a href="a01547.html#a9961967087216e97f76283f29e1be152">__gnu_cxx::hypergl</a></div><div class="ttdeci">long double hypergl(long double __a, long double __b, long double __c, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01350">specfun.h:1350</a></div></div>
<div class="ttc" id="a01497_html_gada763419b0e21b38e38daa8b6eb56a8c"><div class="ttname"><a href="a01497.html#gada763419b0e21b38e38daa8b6eb56a8c">std::laguerref</a></div><div class="ttdeci">float laguerref(unsigned int __n, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00933">specfun.h:933</a></div></div>
<div class="ttc" id="a01497_html_gacae65579b397fddcd27954380d364a58"><div class="ttname"><a href="a01497.html#gacae65579b397fddcd27954380d364a58">std::laguerre</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type laguerre(unsigned int __n, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00962">specfun.h:962</a></div></div>
<div class="ttc" id="a01497_html_ga22fcc678829f0daf2de257896378e7e0"><div class="ttname"><a href="a01497.html#ga22fcc678829f0daf2de257896378e7e0">std::comp_ellint_2</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type comp_ellint_2(_Tp __k)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00438">specfun.h:438</a></div></div>
<div class="ttc" id="a01497_html_gaaf8b141edf9163b37ea4f2ed3e0191fc"><div class="ttname"><a href="a01497.html#gaaf8b141edf9163b37ea4f2ed3e0191fc">std::laguerrel</a></div><div class="ttdeci">long double laguerrel(unsigned int __n, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00943">specfun.h:943</a></div></div>
<div class="ttc" id="a01547_html_af317ba724c44b3a8271fe341d9870173"><div class="ttname"><a href="a01547.html#af317ba724c44b3a8271fe341d9870173">__gnu_cxx::airy_aif</a></div><div class="ttdeci">float airy_aif(float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01215">specfun.h:1215</a></div></div>
<div class="ttc" id="a01547_html_af52cf49736c63b0bb000be98b53c221f"><div class="ttname"><a href="a01547.html#af52cf49736c63b0bb000be98b53c221f">__gnu_cxx::hyperg</a></div><div class="ttdeci">__gnu_cxx::__promote_4< _Tpa, _Tpb, _Tpc, _Tp >::__type hyperg(_Tpa __a, _Tpb __b, _Tpc __c, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01372">specfun.h:1372</a></div></div>
<div class="ttc" id="a01497_html_gaaf738427d4da0bda66bc2274dfb853a7"><div class="ttname"><a href="a01497.html#gaaf738427d4da0bda66bc2274dfb853a7">std::cyl_bessel_if</a></div><div class="ttdeci">float cyl_bessel_if(float __nu, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00504">specfun.h:504</a></div></div>
<div class="ttc" id="a01497_html_ga795383fa51e02351000b410b478d824f"><div class="ttname"><a href="a01497.html#ga795383fa51e02351000b410b478d824f">std::ellint_1l</a></div><div class="ttdeci">long double ellint_1l(long double __k, long double __phi)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00706">specfun.h:706</a></div></div>
<div class="ttc" id="a01497_html_gade8e94a80520a8b628b2d658755b25c0"><div class="ttname"><a href="a01497.html#gade8e94a80520a8b628b2d658755b25c0">std::cyl_bessel_jl</a></div><div class="ttdeci">long double cyl_bessel_jl(long double __nu, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00560">specfun.h:560</a></div></div>
<div class="ttc" id="a01497_html_ga1e92da3b878d75270f38d3ec9b513086"><div class="ttname"><a href="a01497.html#ga1e92da3b878d75270f38d3ec9b513086">std::riemann_zetal</a></div><div class="ttdeci">long double riemann_zetal(long double __s)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01032">specfun.h:1032</a></div></div>
<div class="ttc" id="a01497_html_gae6b3df5556f38a7d72f9b4457d856f9c"><div class="ttname"><a href="a01497.html#gae6b3df5556f38a7d72f9b4457d856f9c">std::ellint_1</a></div><div class="ttdeci">__gnu_cxx::__promote_2< _Tp, _Tpp >::__type ellint_1(_Tp __k, _Tpp __phi)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00729">specfun.h:729</a></div></div>
<div class="ttc" id="a01497_html_ga1a80bd2c15bc9fbecda2630a9e9409e7"><div class="ttname"><a href="a01497.html#ga1a80bd2c15bc9fbecda2630a9e9409e7">std::ellint_3f</a></div><div class="ttdeci">float ellint_3f(float __k, float __nu, float __phi)</div><div class="ttdoc">Return the incomplete elliptic integral of the third kind for float argument.</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00792">specfun.h:792</a></div></div>
<div class="ttc" id="a01497_html_ga7247d3dd77c1ff5df3c059fed862dc48"><div class="ttname"><a href="a01497.html#ga7247d3dd77c1ff5df3c059fed862dc48">std::comp_ellint_1l</a></div><div class="ttdeci">long double comp_ellint_1l(long double __k)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00368">specfun.h:368</a></div></div>
<div class="ttc" id="a01497_html_gad833404645e24b7f0598a640ff92d623"><div class="ttname"><a href="a01497.html#gad833404645e24b7f0598a640ff92d623">std::comp_ellint_3</a></div><div class="ttdeci">__gnu_cxx::__promote_2< _Tp, _Tpn >::__type comp_ellint_3(_Tp __k, _Tpn __nu)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00489">specfun.h:489</a></div></div>
<div class="ttc" id="a01497_html_ga5b7c72ab85e361cbd73f1a3b5f0725a6"><div class="ttname"><a href="a01497.html#ga5b7c72ab85e361cbd73f1a3b5f0725a6">std::cyl_neumann</a></div><div class="ttdeci">__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type cyl_neumann(_Tpnu __nu, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00681">specfun.h:681</a></div></div>
<div class="ttc" id="a01497_html_ga1f50047f9aab0ec8b1a1615fe9fbe32f"><div class="ttname"><a href="a01497.html#ga1f50047f9aab0ec8b1a1615fe9fbe32f">std::cyl_bessel_kf</a></div><div class="ttdeci">float cyl_bessel_kf(float __nu, float __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00596">specfun.h:596</a></div></div>
<div class="ttc" id="a01497_html_ga1c9b5a5c36f000a4f0a55f7fcc486cb0"><div class="ttname"><a href="a01497.html#ga1c9b5a5c36f000a4f0a55f7fcc486cb0">std::cyl_bessel_i</a></div><div class="ttdeci">__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type cyl_bessel_i(_Tpnu __nu, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00535">specfun.h:535</a></div></div>
<div class="ttc" id="a01497_html_gaf6eac7fcb98e25b8f3f7d1b95fa7add8"><div class="ttname"><a href="a01497.html#gaf6eac7fcb98e25b8f3f7d1b95fa7add8">std::legendre</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type legendre(unsigned int __l, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01007">specfun.h:1007</a></div></div>
<div class="ttc" id="a01497_html_gac35194b926270d7857d651e06198c7d3"><div class="ttname"><a href="a01497.html#gac35194b926270d7857d651e06198c7d3">std::cyl_bessel_kl</a></div><div class="ttdeci">long double cyl_bessel_kl(long double __nu, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00606">specfun.h:606</a></div></div>
<div class="ttc" id="a01497_html_ga67a6bfed9b6ab692e8c798b674431424"><div class="ttname"><a href="a01497.html#ga67a6bfed9b6ab692e8c798b674431424">std::riemann_zeta</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type riemann_zeta(_Tp __s)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01058">specfun.h:1058</a></div></div>
<div class="ttc" id="a01497_html_gaae635d28c06a3be2679901b382090852"><div class="ttname"><a href="a01497.html#gaae635d28c06a3be2679901b382090852">std::sph_legendref</a></div><div class="ttdeci">float sph_legendref(unsigned int __l, unsigned int __m, float __theta)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01117">specfun.h:1117</a></div></div>
<div class="ttc" id="a01497_html_gac8e245671fb2df5de5fd978d03081f6c"><div class="ttname"><a href="a01497.html#gac8e245671fb2df5de5fd978d03081f6c">std::assoc_laguerrel</a></div><div class="ttdeci">long double assoc_laguerrel(unsigned int __n, unsigned int __m, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00216">specfun.h:216</a></div></div>
<div class="ttc" id="a01497_html_gaf8986bae9a523c48d861d233835bda8f"><div class="ttname"><a href="a01497.html#gaf8986bae9a523c48d861d233835bda8f">std::cyl_neumannl</a></div><div class="ttdeci">long double cyl_neumannl(long double __nu, long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00658">specfun.h:658</a></div></div>
<div class="ttc" id="a01497_html_ga6a7220c87c942db48b18b527d92bbd2d"><div class="ttname"><a href="a01497.html#ga6a7220c87c942db48b18b527d92bbd2d">std::beta</a></div><div class="ttdeci">__gnu_cxx::__promote_2< _Tpa, _Tpb >::__type beta(_Tpa __a, _Tpb __b)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00343">specfun.h:343</a></div></div>
<div class="ttc" id="a01497_html_ga1cf4362a67ab5bae9e6b69c867e85371"><div class="ttname"><a href="a01497.html#ga1cf4362a67ab5bae9e6b69c867e85371">std::sph_neumann</a></div><div class="ttdeci">__gnu_cxx::__promote< _Tp >::__type sph_neumann(unsigned int __n, _Tp __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01193">specfun.h:1193</a></div></div>
<div class="ttc" id="a01547_html_a59240b3f40177e5187f3f194f624f0f8"><div class="ttname"><a href="a01547.html#a59240b3f40177e5187f3f194f624f0f8">__gnu_cxx::airy_bil</a></div><div class="ttdeci">long double airy_bil(long double __x)</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l01261">specfun.h:1261</a></div></div>
<div class="ttc" id="a01497_html_ga1ca081fee102cd0d4d6b091285e495e5"><div class="ttname"><a href="a01497.html#ga1ca081fee102cd0d4d6b091285e495e5">std::comp_ellint_3l</a></div><div class="ttdeci">long double comp_ellint_3l(long double __k, long double __nu)</div><div class="ttdoc">Return the complete elliptic integral of the third kind for long double modulus k.</div><div class="ttdef"><b>Definition:</b> <a href="a00512_source.html#l00463">specfun.h:463</a></div></div>
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