// Mathematical Special Functions for -*- C++ -*- // Copyright (C) 2006-2020 Free Software Foundation, Inc. // // This file is part of the GNU ISO C++ Library. This library is free // software; you can redistribute it and/or modify it under the // terms of the GNU General Public License as published by the // Free Software Foundation; either version 3, or (at your option) // any later version. // This library is distributed in the hope that it will be useful, // but WITHOUT ANY WARRANTY; without even the implied warranty of // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the // GNU General Public License for more details. // Under Section 7 of GPL version 3, you are granted additional // permissions described in the GCC Runtime Library Exception, version // 3.1, as published by the Free Software Foundation. // You should have received a copy of the GNU General Public License and // a copy of the GCC Runtime Library Exception along with this program; // see the files COPYING3 and COPYING.RUNTIME respectively. If not, see // <http://www.gnu.org/licenses/>. /** @file bits/specfun.h * This is an internal header file, included by other library headers. * Do not attempt to use it directly. @headername{cmath} */ #ifndef _GLIBCXX_BITS_SPECFUN_H #define _GLIBCXX_BITS_SPECFUN_H 1 #pragma GCC visibility push(default) #include <bits/c++config.h> #define __STDCPP_MATH_SPEC_FUNCS__ 201003L #define __cpp_lib_math_special_functions 201603L #if __cplusplus <= 201403L && __STDCPP_WANT_MATH_SPEC_FUNCS__ == 0 # error include <cmath> and define __STDCPP_WANT_MATH_SPEC_FUNCS__ #endif #include <bits/stl_algobase.h> #include <limits> #include <type_traits> #include <tr1/gamma.tcc> #include <tr1/bessel_function.tcc> #include <tr1/beta_function.tcc> #include <tr1/ell_integral.tcc> #include <tr1/exp_integral.tcc> #include <tr1/hypergeometric.tcc> #include <tr1/legendre_function.tcc> #include <tr1/modified_bessel_func.tcc> #include <tr1/poly_hermite.tcc> #include <tr1/poly_laguerre.tcc> #include <tr1/riemann_zeta.tcc> namespace std _GLIBCXX_VISIBILITY(default) { _GLIBCXX_BEGIN_NAMESPACE_VERSION /** * @defgroup mathsf Mathematical Special Functions * @ingroup numerics * * @section mathsf_desc Mathematical Special Functions * * A collection of advanced mathematical special functions, * defined by ISO/IEC IS 29124 and then added to ISO C++ 2017. * * * @subsection mathsf_intro Introduction and History * The first significant library upgrade on the road to C++2011, * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2005/n1836.pdf"> * TR1</a>, included a set of 23 mathematical functions that significantly * extended the standard transcendental functions inherited from C and declared * in @<cmath@>. * * Although most components from TR1 were eventually adopted for C++11 these * math functions were left behind out of concern for implementability. * The math functions were published as a separate international standard * <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2010/n3060.pdf"> * IS 29124 - Extensions to the C++ Library to Support Mathematical Special * Functions</a>. * * For C++17 these functions were incorporated into the main standard. * * @subsection mathsf_contents Contents * The following functions are implemented in namespace @c std: * - @ref assoc_laguerre "assoc_laguerre - Associated Laguerre functions" * - @ref assoc_legendre "assoc_legendre - Associated Legendre functions" * - @ref beta "beta - Beta functions" * - @ref comp_ellint_1 "comp_ellint_1 - Complete elliptic functions of the first kind" * - @ref comp_ellint_2 "comp_ellint_2 - Complete elliptic functions of the second kind" * - @ref comp_ellint_3 "comp_ellint_3 - Complete elliptic functions of the third kind" * - @ref cyl_bessel_i "cyl_bessel_i - Regular modified cylindrical Bessel functions" * - @ref cyl_bessel_j "cyl_bessel_j - Cylindrical Bessel functions of the first kind" * - @ref cyl_bessel_k "cyl_bessel_k - Irregular modified cylindrical Bessel functions" * - @ref cyl_neumann "cyl_neumann - Cylindrical Neumann functions or Cylindrical Bessel functions of the second kind" * - @ref ellint_1 "ellint_1 - Incomplete elliptic functions of the first kind" * - @ref ellint_2 "ellint_2 - Incomplete elliptic functions of the second kind" * - @ref ellint_3 "ellint_3 - Incomplete elliptic functions of the third kind" * - @ref expint "expint - The exponential integral" * - @ref hermite "hermite - Hermite polynomials" * - @ref laguerre "laguerre - Laguerre functions" * - @ref legendre "legendre - Legendre polynomials" * - @ref riemann_zeta "riemann_zeta - The Riemann zeta function" * - @ref sph_bessel "sph_bessel - Spherical Bessel functions" * - @ref sph_legendre "sph_legendre - Spherical Legendre functions" * - @ref sph_neumann "sph_neumann - Spherical Neumann functions" * * The hypergeometric functions were stricken from the TR29124 and C++17 * versions of this math library because of implementation concerns. * However, since they were in the TR1 version and since they are popular * we kept them as an extension in namespace @c __gnu_cxx: * - @ref __gnu_cxx::conf_hyperg "conf_hyperg - Confluent hypergeometric functions" * - @ref __gnu_cxx::hyperg "hyperg - Hypergeometric functions" * * <!-- @subsection mathsf_general General Features --> * * @subsection mathsf_promotion Argument Promotion * The arguments suppled to the non-suffixed functions will be promoted * according to the following rules: * 1. If any argument intended to be floating point is given an integral value * That integral value is promoted to double. * 2. All floating point arguments are promoted up to the largest floating * point precision among them. * * @subsection mathsf_NaN NaN Arguments * If any of the floating point arguments supplied to these functions is * invalid or NaN (std::numeric_limits<Tp>::quiet_NaN), * the value NaN is returned. * * @subsection mathsf_impl Implementation * * We strive to implement the underlying math with type generic algorithms * to the greatest extent possible. In practice, the functions are thin * wrappers that dispatch to function templates. Type dependence is * controlled with std::numeric_limits and functions thereof. * * We don't promote @c float to @c double or @c double to <tt>long double</tt> * reflexively. The goal is for @c float functions to operate more quickly, * at the cost of @c float accuracy and possibly a smaller domain of validity. * Similaryly, <tt>long double</tt> should give you more dynamic range * and slightly more pecision than @c double on many systems. * * @subsection mathsf_testing Testing * * These functions have been tested against equivalent implementations * from the <a href="http://www.gnu.org/software/gsl"> * Gnu Scientific Library, GSL</a> and * <a href="http://www.boost.org/doc/libs/1_60_0/libs/math/doc/html/index.html">Boost</a> * and the ratio * @f[ * \frac{|f - f_{test}|}{|f_{test}|} * @f] * is generally found to be within 10<sup>-15</sup> for 64-bit double on * linux-x86_64 systems over most of the ranges of validity. * * @todo Provide accuracy comparisons on a per-function basis for a small * number of targets. * * @subsection mathsf_bibliography General Bibliography * * @see Abramowitz and Stegun: Handbook of Mathematical Functions, * with Formulas, Graphs, and Mathematical Tables * Edited by Milton Abramowitz and Irene A. Stegun, * National Bureau of Standards Applied Mathematics Series - 55 * Issued June 1964, Tenth Printing, December 1972, with corrections * Electronic versions of A&S abound including both pdf and navigable html. * @see for example http://people.math.sfu.ca/~cbm/aands/ * * @see The old A&S has been redone as the * NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/ * This version is far more navigable and includes more recent work. * * @see An Atlas of Functions: with Equator, the Atlas Function Calculator * 2nd Edition, by Oldham, Keith B., Myland, Jan, Spanier, Jerome * * @see Asymptotics and Special Functions by Frank W. J. Olver, * Academic Press, 1974 * * @see Numerical Recipes in C, The Art of Scientific Computing, * by William H. Press, Second Ed., Saul A. Teukolsky, * William T. Vetterling, and Brian P. Flannery, * Cambridge University Press, 1992 * * @see The Special Functions and Their Approximations: Volumes 1 and 2, * by Yudell L. Luke, Academic Press, 1969 * * @{ */ // Associated Laguerre polynomials /** * Return the associated Laguerre polynomial of order @c n, * degree @c m: @f$ L_n^m(x) @f$ for @c float argument. * * @see assoc_laguerre for more details. */ inline float assoc_laguerref(unsigned int __n, unsigned int __m, float __x) { return __detail::__assoc_laguerre<float>(__n, __m, __x); } /** * Return the associated Laguerre polynomial of order @c n, * degree @c m: @f$ L_n^m(x) @f$. * * @see assoc_laguerre for more details. */ inline long double assoc_laguerrel(unsigned int __n, unsigned int __m, long double __x) { return __detail::__assoc_laguerre<long double>(__n, __m, __x); } /** * Return the associated Laguerre polynomial of nonnegative order @c n, * nonnegative degree @c m and real argument @c x: @f$ L_n^m(x) @f$. * * The associated Laguerre function of real degree @f$ \alpha @f$, * @f$ L_n^\alpha(x) @f$, is defined by * @f[ * L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!} * {}_1F_1(-n; \alpha + 1; x) * @f] * where @f$ (\alpha)_n @f$ is the Pochhammer symbol and * @f$ {}_1F_1(a; c; x) @f$ is the confluent hypergeometric function. * * The associated Laguerre polynomial is defined for integral * degree @f$ \alpha = m @f$ by: * @f[ * L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x) * @f] * where the Laguerre polynomial is defined by: * @f[ * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) * @f] * and @f$ x >= 0 @f$. * @see laguerre for details of the Laguerre function of degree @c n * * @tparam _Tp The floating-point type of the argument @c __x. * @param __n The order of the Laguerre function, <tt>__n >= 0</tt>. * @param __m The degree of the Laguerre function, <tt>__m >= 0</tt>. * @param __x The argument of the Laguerre function, <tt>__x >= 0</tt>. * @throw std::domain_error if <tt>__x < 0</tt>. */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__assoc_laguerre<__type>(__n, __m, __x); } // Associated Legendre functions /** * Return the associated Legendre function of degree @c l and order @c m * for @c float argument. * * @see assoc_legendre for more details. */ inline float assoc_legendref(unsigned int __l, unsigned int __m, float __x) { return __detail::__assoc_legendre_p<float>(__l, __m, __x); } /** * Return the associated Legendre function of degree @c l and order @c m. * * @see assoc_legendre for more details. */ inline long double assoc_legendrel(unsigned int __l, unsigned int __m, long double __x) { return __detail::__assoc_legendre_p<long double>(__l, __m, __x); } /** * Return the associated Legendre function of degree @c l and order @c m. * * The associated Legendre function is derived from the Legendre function * @f$ P_l(x) @f$ by the Rodrigues formula: * @f[ * P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x) * @f] * @see legendre for details of the Legendre function of degree @c l * * @tparam _Tp The floating-point type of the argument @c __x. * @param __l The degree <tt>__l >= 0</tt>. * @param __m The order <tt>__m <= l</tt>. * @param __x The argument, <tt>abs(__x) <= 1</tt>. * @throw std::domain_error if <tt>abs(__x) > 1</tt>. */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type assoc_legendre(unsigned int __l, unsigned int __m, _Tp __x) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__assoc_legendre_p<__type>(__l, __m, __x); } // Beta functions /** * Return the beta function, @f$ B(a,b) @f$, for @c float parameters @c a, @c b. * * @see beta for more details. */ inline float betaf(float __a, float __b) { return __detail::__beta<float>(__a, __b); } /** * Return the beta function, @f$B(a,b)@f$, for long double * parameters @c a, @c b. * * @see beta for more details. */ inline long double betal(long double __a, long double __b) { return __detail::__beta<long double>(__a, __b); } /** * Return the beta function, @f$B(a,b)@f$, for real parameters @c a, @c b. * * The beta function is defined by * @f[ * B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt * = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} * @f] * where @f$ a > 0 @f$ and @f$ b > 0 @f$ * * @tparam _Tpa The floating-point type of the parameter @c __a. * @tparam _Tpb The floating-point type of the parameter @c __b. * @param __a The first argument of the beta function, <tt> __a > 0 </tt>. * @param __b The second argument of the beta function, <tt> __b > 0 </tt>. * @throw std::domain_error if <tt> __a < 0 </tt> or <tt> __b < 0 </tt>. */ template<typename _Tpa, typename _Tpb> inline typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type beta(_Tpa __a, _Tpb __b) { typedef typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type __type; return __detail::__beta<__type>(__a, __b); } // Complete elliptic integrals of the first kind /** * Return the complete elliptic integral of the first kind @f$ E(k) @f$ * for @c float modulus @c k. * * @see comp_ellint_1 for details. */ inline float comp_ellint_1f(float __k) { return __detail::__comp_ellint_1<float>(__k); } /** * Return the complete elliptic integral of the first kind @f$ E(k) @f$ * for long double modulus @c k. * * @see comp_ellint_1 for details. */ inline long double comp_ellint_1l(long double __k) { return __detail::__comp_ellint_1<long double>(__k); } /** * Return the complete elliptic integral of the first kind * @f$ K(k) @f$ for real modulus @c k. * * The complete elliptic integral of the first kind is defined as * @f[ * K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta} * {\sqrt{1 - k^2 sin^2\theta}} * @f] * where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the * first kind and the modulus @f$ |k| <= 1 @f$. * @see ellint_1 for details of the incomplete elliptic function * of the first kind. * * @tparam _Tp The floating-point type of the modulus @c __k. * @param __k The modulus, <tt> abs(__k) <= 1 </tt> * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type comp_ellint_1(_Tp __k) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__comp_ellint_1<__type>(__k); } // Complete elliptic integrals of the second kind /** * Return the complete elliptic integral of the second kind @f$ E(k) @f$ * for @c float modulus @c k. * * @see comp_ellint_2 for details. */ inline float comp_ellint_2f(float __k) { return __detail::__comp_ellint_2<float>(__k); } /** * Return the complete elliptic integral of the second kind @f$ E(k) @f$ * for long double modulus @c k. * * @see comp_ellint_2 for details. */ inline long double comp_ellint_2l(long double __k) { return __detail::__comp_ellint_2<long double>(__k); } /** * Return the complete elliptic integral of the second kind @f$ E(k) @f$ * for real modulus @c k. * * The complete elliptic integral of the second kind is defined as * @f[ * E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta} * @f] * where @f$ E(k,\phi) @f$ is the incomplete elliptic integral of the * second kind and the modulus @f$ |k| <= 1 @f$. * @see ellint_2 for details of the incomplete elliptic function * of the second kind. * * @tparam _Tp The floating-point type of the modulus @c __k. * @param __k The modulus, @c abs(__k) <= 1 * @throw std::domain_error if @c abs(__k) > 1. */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type comp_ellint_2(_Tp __k) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__comp_ellint_2<__type>(__k); } // Complete elliptic integrals of the third kind /** * @brief Return the complete elliptic integral of the third kind * @f$ \Pi(k,\nu) @f$ for @c float modulus @c k. * * @see comp_ellint_3 for details. */ inline float comp_ellint_3f(float __k, float __nu) { return __detail::__comp_ellint_3<float>(__k, __nu); } /** * @brief Return the complete elliptic integral of the third kind * @f$ \Pi(k,\nu) @f$ for <tt>long double</tt> modulus @c k. * * @see comp_ellint_3 for details. */ inline long double comp_ellint_3l(long double __k, long double __nu) { return __detail::__comp_ellint_3<long double>(__k, __nu); } /** * Return the complete elliptic integral of the third kind * @f$ \Pi(k,\nu) = \Pi(k,\nu,\pi/2) @f$ for real modulus @c k. * * The complete elliptic integral of the third kind is defined as * @f[ * \Pi(k,\nu) = \Pi(k,\nu,\pi/2) = \int_0^{\pi/2} * \frac{d\theta} * {(1 - \nu \sin^2\theta)\sqrt{1 - k^2 \sin^2\theta}} * @f] * where @f$ \Pi(k,\nu,\phi) @f$ is the incomplete elliptic integral of the * second kind and the modulus @f$ |k| <= 1 @f$. * @see ellint_3 for details of the incomplete elliptic function * of the third kind. * * @tparam _Tp The floating-point type of the modulus @c __k. * @tparam _Tpn The floating-point type of the argument @c __nu. * @param __k The modulus, @c abs(__k) <= 1 * @param __nu The argument * @throw std::domain_error if @c abs(__k) > 1. */ template<typename _Tp, typename _Tpn> inline typename __gnu_cxx::__promote_2<_Tp, _Tpn>::__type comp_ellint_3(_Tp __k, _Tpn __nu) { typedef typename __gnu_cxx::__promote_2<_Tp, _Tpn>::__type __type; return __detail::__comp_ellint_3<__type>(__k, __nu); } // Regular modified cylindrical Bessel functions /** * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. * * @see cyl_bessel_i for setails. */ inline float cyl_bessel_if(float __nu, float __x) { return __detail::__cyl_bessel_i<float>(__nu, __x); } /** * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. * * @see cyl_bessel_i for setails. */ inline long double cyl_bessel_il(long double __nu, long double __x) { return __detail::__cyl_bessel_i<long double>(__nu, __x); } /** * Return the regular modified Bessel function @f$ I_{\nu}(x) @f$ * for real order @f$ \nu @f$ and argument @f$ x >= 0 @f$. * * The regular modified cylindrical Bessel function is: * @f[ * I_{\nu}(x) = i^{-\nu}J_\nu(ix) = \sum_{k=0}^{\infty} * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} * @f] * * @tparam _Tpnu The floating-point type of the order @c __nu. * @tparam _Tp The floating-point type of the argument @c __x. * @param __nu The order * @param __x The argument, <tt> __x >= 0 </tt> * @throw std::domain_error if <tt> __x < 0 </tt>. */ template<typename _Tpnu, typename _Tp> inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type cyl_bessel_i(_Tpnu __nu, _Tp __x) { typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; return __detail::__cyl_bessel_i<__type>(__nu, __x); } // Cylindrical Bessel functions (of the first kind) /** * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$ * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. * * @see cyl_bessel_j for setails. */ inline float cyl_bessel_jf(float __nu, float __x) { return __detail::__cyl_bessel_j<float>(__nu, __x); } /** * Return the Bessel function of the first kind @f$ J_{\nu}(x) @f$ * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. * * @see cyl_bessel_j for setails. */ inline long double cyl_bessel_jl(long double __nu, long double __x) { return __detail::__cyl_bessel_j<long double>(__nu, __x); } /** * Return the Bessel function @f$ J_{\nu}(x) @f$ of real order @f$ \nu @f$ * and argument @f$ x >= 0 @f$. * * The cylindrical Bessel function is: * @f[ * J_{\nu}(x) = \sum_{k=0}^{\infty} * \frac{(-1)^k (x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} * @f] * * @tparam _Tpnu The floating-point type of the order @c __nu. * @tparam _Tp The floating-point type of the argument @c __x. * @param __nu The order * @param __x The argument, <tt> __x >= 0 </tt> * @throw std::domain_error if <tt> __x < 0 </tt>. */ template<typename _Tpnu, typename _Tp> inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type cyl_bessel_j(_Tpnu __nu, _Tp __x) { typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; return __detail::__cyl_bessel_j<__type>(__nu, __x); } // Irregular modified cylindrical Bessel functions /** * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ * for @c float order @f$ \nu @f$ and argument @f$ x >= 0 @f$. * * @see cyl_bessel_k for setails. */ inline float cyl_bessel_kf(float __nu, float __x) { return __detail::__cyl_bessel_k<float>(__nu, __x); } /** * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ * for <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x >= 0 @f$. * * @see cyl_bessel_k for setails. */ inline long double cyl_bessel_kl(long double __nu, long double __x) { return __detail::__cyl_bessel_k<long double>(__nu, __x); } /** * Return the irregular modified Bessel function @f$ K_{\nu}(x) @f$ * of real order @f$ \nu @f$ and argument @f$ x @f$. * * The irregular modified Bessel function is defined by: * @f[ * K_{\nu}(x) = \frac{\pi}{2} * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} * @f] * where for integral @f$ \nu = n @f$ a limit is taken: * @f$ lim_{\nu \to n} @f$. * For negative argument we have simply: * @f[ * K_{-\nu}(x) = K_{\nu}(x) * @f] * * @tparam _Tpnu The floating-point type of the order @c __nu. * @tparam _Tp The floating-point type of the argument @c __x. * @param __nu The order * @param __x The argument, <tt> __x >= 0 </tt> * @throw std::domain_error if <tt> __x < 0 </tt>. */ template<typename _Tpnu, typename _Tp> inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type cyl_bessel_k(_Tpnu __nu, _Tp __x) { typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; return __detail::__cyl_bessel_k<__type>(__nu, __x); } // Cylindrical Neumann functions /** * Return the Neumann function @f$ N_{\nu}(x) @f$ * of @c float order @f$ \nu @f$ and argument @f$ x @f$. * * @see cyl_neumann for setails. */ inline float cyl_neumannf(float __nu, float __x) { return __detail::__cyl_neumann_n<float>(__nu, __x); } /** * Return the Neumann function @f$ N_{\nu}(x) @f$ * of <tt>long double</tt> order @f$ \nu @f$ and argument @f$ x @f$. * * @see cyl_neumann for setails. */ inline long double cyl_neumannl(long double __nu, long double __x) { return __detail::__cyl_neumann_n<long double>(__nu, __x); } /** * Return the Neumann function @f$ N_{\nu}(x) @f$ * of real order @f$ \nu @f$ and argument @f$ x >= 0 @f$. * * The Neumann function is defined by: * @f[ * N_{\nu}(x) = \frac{J_{\nu}(x) \cos \nu\pi - J_{-\nu}(x)} * {\sin \nu\pi} * @f] * where @f$ x >= 0 @f$ and for integral order @f$ \nu = n @f$ * a limit is taken: @f$ lim_{\nu \to n} @f$. * * @tparam _Tpnu The floating-point type of the order @c __nu. * @tparam _Tp The floating-point type of the argument @c __x. * @param __nu The order * @param __x The argument, <tt> __x >= 0 </tt> * @throw std::domain_error if <tt> __x < 0 </tt>. */ template<typename _Tpnu, typename _Tp> inline typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type cyl_neumann(_Tpnu __nu, _Tp __x) { typedef typename __gnu_cxx::__promote_2<_Tpnu, _Tp>::__type __type; return __detail::__cyl_neumann_n<__type>(__nu, __x); } // Incomplete elliptic integrals of the first kind /** * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$ * for @c float modulus @f$ k @f$ and angle @f$ \phi @f$. * * @see ellint_1 for details. */ inline float ellint_1f(float __k, float __phi) { return __detail::__ellint_1<float>(__k, __phi); } /** * Return the incomplete elliptic integral of the first kind @f$ E(k,\phi) @f$ * for <tt>long double</tt> modulus @f$ k @f$ and angle @f$ \phi @f$. * * @see ellint_1 for details. */ inline long double ellint_1l(long double __k, long double __phi) { return __detail::__ellint_1<long double>(__k, __phi); } /** * Return the incomplete elliptic integral of the first kind @f$ F(k,\phi) @f$ * for @c real modulus @f$ k @f$ and angle @f$ \phi @f$. * * The incomplete elliptic integral of the first kind is defined as * @f[ * F(k,\phi) = \int_0^{\phi}\frac{d\theta} * {\sqrt{1 - k^2 sin^2\theta}} * @f] * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of * the first kind, @f$ K(k) @f$. @see comp_ellint_1. * * @tparam _Tp The floating-point type of the modulus @c __k. * @tparam _Tpp The floating-point type of the angle @c __phi. * @param __k The modulus, <tt> abs(__k) <= 1 </tt> * @param __phi The integral limit argument in radians * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. */ template<typename _Tp, typename _Tpp> inline typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type ellint_1(_Tp __k, _Tpp __phi) { typedef typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type __type; return __detail::__ellint_1<__type>(__k, __phi); } // Incomplete elliptic integrals of the second kind /** * @brief Return the incomplete elliptic integral of the second kind * @f$ E(k,\phi) @f$ for @c float argument. * * @see ellint_2 for details. */ inline float ellint_2f(float __k, float __phi) { return __detail::__ellint_2<float>(__k, __phi); } /** * @brief Return the incomplete elliptic integral of the second kind * @f$ E(k,\phi) @f$. * * @see ellint_2 for details. */ inline long double ellint_2l(long double __k, long double __phi) { return __detail::__ellint_2<long double>(__k, __phi); } /** * Return the incomplete elliptic integral of the second kind * @f$ E(k,\phi) @f$. * * The incomplete elliptic integral of the second kind is defined as * @f[ * E(k,\phi) = \int_0^{\phi} \sqrt{1 - k^2 sin^2\theta} * @f] * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of * the second kind, @f$ E(k) @f$. @see comp_ellint_2. * * @tparam _Tp The floating-point type of the modulus @c __k. * @tparam _Tpp The floating-point type of the angle @c __phi. * @param __k The modulus, <tt> abs(__k) <= 1 </tt> * @param __phi The integral limit argument in radians * @return The elliptic function of the second kind. * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. */ template<typename _Tp, typename _Tpp> inline typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type ellint_2(_Tp __k, _Tpp __phi) { typedef typename __gnu_cxx::__promote_2<_Tp, _Tpp>::__type __type; return __detail::__ellint_2<__type>(__k, __phi); } // Incomplete elliptic integrals of the third kind /** * @brief Return the incomplete elliptic integral of the third kind * @f$ \Pi(k,\nu,\phi) @f$ for @c float argument. * * @see ellint_3 for details. */ inline float ellint_3f(float __k, float __nu, float __phi) { return __detail::__ellint_3<float>(__k, __nu, __phi); } /** * @brief Return the incomplete elliptic integral of the third kind * @f$ \Pi(k,\nu,\phi) @f$. * * @see ellint_3 for details. */ inline long double ellint_3l(long double __k, long double __nu, long double __phi) { return __detail::__ellint_3<long double>(__k, __nu, __phi); } /** * @brief Return the incomplete elliptic integral of the third kind * @f$ \Pi(k,\nu,\phi) @f$. * * The incomplete elliptic integral of the third kind is defined by: * @f[ * \Pi(k,\nu,\phi) = \int_0^{\phi} * \frac{d\theta} * {(1 - \nu \sin^2\theta) * \sqrt{1 - k^2 \sin^2\theta}} * @f] * For @f$ \phi= \pi/2 @f$ this becomes the complete elliptic integral of * the third kind, @f$ \Pi(k,\nu) @f$. @see comp_ellint_3. * * @tparam _Tp The floating-point type of the modulus @c __k. * @tparam _Tpn The floating-point type of the argument @c __nu. * @tparam _Tpp The floating-point type of the angle @c __phi. * @param __k The modulus, <tt> abs(__k) <= 1 </tt> * @param __nu The second argument * @param __phi The integral limit argument in radians * @return The elliptic function of the third kind. * @throw std::domain_error if <tt> abs(__k) > 1 </tt>. */ template<typename _Tp, typename _Tpn, typename _Tpp> inline typename __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type ellint_3(_Tp __k, _Tpn __nu, _Tpp __phi) { typedef typename __gnu_cxx::__promote_3<_Tp, _Tpn, _Tpp>::__type __type; return __detail::__ellint_3<__type>(__k, __nu, __phi); } // Exponential integrals /** * Return the exponential integral @f$ Ei(x) @f$ for @c float argument @c x. * * @see expint for details. */ inline float expintf(float __x) { return __detail::__expint<float>(__x); } /** * Return the exponential integral @f$ Ei(x) @f$ * for <tt>long double</tt> argument @c x. * * @see expint for details. */ inline long double expintl(long double __x) { return __detail::__expint<long double>(__x); } /** * Return the exponential integral @f$ Ei(x) @f$ for @c real argument @c x. * * The exponential integral is given by * \f[ * Ei(x) = -\int_{-x}^\infty \frac{e^t}{t} dt * \f] * * @tparam _Tp The floating-point type of the argument @c __x. * @param __x The argument of the exponential integral function. */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type expint(_Tp __x) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__expint<__type>(__x); } // Hermite polynomials /** * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n * and float argument @c x. * * @see hermite for details. */ inline float hermitef(unsigned int __n, float __x) { return __detail::__poly_hermite<float>(__n, __x); } /** * Return the Hermite polynomial @f$ H_n(x) @f$ of nonnegative order n * and <tt>long double</tt> argument @c x. * * @see hermite for details. */ inline long double hermitel(unsigned int __n, long double __x) { return __detail::__poly_hermite<long double>(__n, __x); } /** * Return the Hermite polynomial @f$ H_n(x) @f$ of order n * and @c real argument @c x. * * The Hermite polynomial is defined by: * @f[ * H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} * @f] * * The Hermite polynomial obeys a reflection formula: * @f[ * H_n(-x) = (-1)^n H_n(x) * @f] * * @tparam _Tp The floating-point type of the argument @c __x. * @param __n The order * @param __x The argument */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type hermite(unsigned int __n, _Tp __x) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__poly_hermite<__type>(__n, __x); } // Laguerre polynomials /** * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n * and @c float argument @f$ x >= 0 @f$. * * @see laguerre for more details. */ inline float laguerref(unsigned int __n, float __x) { return __detail::__laguerre<float>(__n, __x); } /** * Returns the Laguerre polynomial @f$ L_n(x) @f$ of nonnegative degree @c n * and <tt>long double</tt> argument @f$ x >= 0 @f$. * * @see laguerre for more details. */ inline long double laguerrel(unsigned int __n, long double __x) { return __detail::__laguerre<long double>(__n, __x); } /** * Returns the Laguerre polynomial @f$ L_n(x) @f$ * of nonnegative degree @c n and real argument @f$ x >= 0 @f$. * * The Laguerre polynomial is defined by: * @f[ * L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x}) * @f] * * @tparam _Tp The floating-point type of the argument @c __x. * @param __n The nonnegative order * @param __x The argument <tt> __x >= 0 </tt> * @throw std::domain_error if <tt> __x < 0 </tt>. */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type laguerre(unsigned int __n, _Tp __x) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__laguerre<__type>(__n, __x); } // Legendre polynomials /** * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative * degree @f$ l @f$ and @c float argument @f$ |x| <= 0 @f$. * * @see legendre for more details. */ inline float legendref(unsigned int __l, float __x) { return __detail::__poly_legendre_p<float>(__l, __x); } /** * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative * degree @f$ l @f$ and <tt>long double</tt> argument @f$ |x| <= 0 @f$. * * @see legendre for more details. */ inline long double legendrel(unsigned int __l, long double __x) { return __detail::__poly_legendre_p<long double>(__l, __x); } /** * Return the Legendre polynomial @f$ P_l(x) @f$ of nonnegative * degree @f$ l @f$ and real argument @f$ |x| <= 0 @f$. * * The Legendre function of order @f$ l @f$ and argument @f$ x @f$, * @f$ P_l(x) @f$, is defined by: * @f[ * P_l(x) = \frac{1}{2^l l!}\frac{d^l}{dx^l}(x^2 - 1)^{l} * @f] * * @tparam _Tp The floating-point type of the argument @c __x. * @param __l The degree @f$ l >= 0 @f$ * @param __x The argument @c abs(__x) <= 1 * @throw std::domain_error if @c abs(__x) > 1 */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type legendre(unsigned int __l, _Tp __x) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__poly_legendre_p<__type>(__l, __x); } // Riemann zeta functions /** * Return the Riemann zeta function @f$ \zeta(s) @f$ * for @c float argument @f$ s @f$. * * @see riemann_zeta for more details. */ inline float riemann_zetaf(float __s) { return __detail::__riemann_zeta<float>(__s); } /** * Return the Riemann zeta function @f$ \zeta(s) @f$ * for <tt>long double</tt> argument @f$ s @f$. * * @see riemann_zeta for more details. */ inline long double riemann_zetal(long double __s) { return __detail::__riemann_zeta<long double>(__s); } /** * Return the Riemann zeta function @f$ \zeta(s) @f$ * for real argument @f$ s @f$. * * The Riemann zeta function is defined by: * @f[ * \zeta(s) = \sum_{k=1}^{\infty} k^{-s} \hbox{ for } s > 1 * @f] * and * @f[ * \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{k=1}^{\infty}(-1)^{k-1}k^{-s} * \hbox{ for } 0 <= s <= 1 * @f] * For s < 1 use the reflection formula: * @f[ * \zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s) * @f] * * @tparam _Tp The floating-point type of the argument @c __s. * @param __s The argument <tt> s != 1 </tt> */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type riemann_zeta(_Tp __s) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__riemann_zeta<__type>(__s); } // Spherical Bessel functions /** * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n * and @c float argument @f$ x >= 0 @f$. * * @see sph_bessel for more details. */ inline float sph_besself(unsigned int __n, float __x) { return __detail::__sph_bessel<float>(__n, __x); } /** * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n * and <tt>long double</tt> argument @f$ x >= 0 @f$. * * @see sph_bessel for more details. */ inline long double sph_bessell(unsigned int __n, long double __x) { return __detail::__sph_bessel<long double>(__n, __x); } /** * Return the spherical Bessel function @f$ j_n(x) @f$ of nonnegative order n * and real argument @f$ x >= 0 @f$. * * The spherical Bessel function is defined by: * @f[ * j_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} J_{n+1/2}(x) * @f] * * @tparam _Tp The floating-point type of the argument @c __x. * @param __n The integral order <tt> n >= 0 </tt> * @param __x The real argument <tt> x >= 0 </tt> * @throw std::domain_error if <tt> __x < 0 </tt>. */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type sph_bessel(unsigned int __n, _Tp __x) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__sph_bessel<__type>(__n, __x); } // Spherical associated Legendre functions /** * Return the spherical Legendre function of nonnegative integral * degree @c l and order @c m and float angle @f$ \theta @f$ in radians. * * @see sph_legendre for details. */ inline float sph_legendref(unsigned int __l, unsigned int __m, float __theta) { return __detail::__sph_legendre<float>(__l, __m, __theta); } /** * Return the spherical Legendre function of nonnegative integral * degree @c l and order @c m and <tt>long double</tt> angle @f$ \theta @f$ * in radians. * * @see sph_legendre for details. */ inline long double sph_legendrel(unsigned int __l, unsigned int __m, long double __theta) { return __detail::__sph_legendre<long double>(__l, __m, __theta); } /** * Return the spherical Legendre function of nonnegative integral * degree @c l and order @c m and real angle @f$ \theta @f$ in radians. * * The spherical Legendre function is defined by * @f[ * Y_l^m(\theta,\phi) = (-1)^m[\frac{(2l+1)}{4\pi} * \frac{(l-m)!}{(l+m)!}] * P_l^m(\cos\theta) \exp^{im\phi} * @f] * * @tparam _Tp The floating-point type of the angle @c __theta. * @param __l The order <tt> __l >= 0 </tt> * @param __m The degree <tt> __m >= 0 </tt> and <tt> __m <= __l </tt> * @param __theta The radian polar angle argument */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__sph_legendre<__type>(__l, __m, __theta); } // Spherical Neumann functions /** * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ * and @c float argument @f$ x >= 0 @f$. * * @see sph_neumann for details. */ inline float sph_neumannf(unsigned int __n, float __x) { return __detail::__sph_neumann<float>(__n, __x); } /** * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ * and <tt>long double</tt> @f$ x >= 0 @f$. * * @see sph_neumann for details. */ inline long double sph_neumannl(unsigned int __n, long double __x) { return __detail::__sph_neumann<long double>(__n, __x); } /** * Return the spherical Neumann function of integral order @f$ n >= 0 @f$ * and real argument @f$ x >= 0 @f$. * * The spherical Neumann function is defined by * @f[ * n_n(x) = \left(\frac{\pi}{2x} \right) ^{1/2} N_{n+1/2}(x) * @f] * * @tparam _Tp The floating-point type of the argument @c __x. * @param __n The integral order <tt> n >= 0 </tt> * @param __x The real argument <tt> __x >= 0 </tt> * @throw std::domain_error if <tt> __x < 0 </tt>. */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type sph_neumann(unsigned int __n, _Tp __x) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; return __detail::__sph_neumann<__type>(__n, __x); } // @} group mathsf _GLIBCXX_END_NAMESPACE_VERSION } // namespace std #ifndef __STRICT_ANSI__ namespace __gnu_cxx _GLIBCXX_VISIBILITY(default) { _GLIBCXX_BEGIN_NAMESPACE_VERSION /** @addtogroup mathsf * @{ */ // Airy functions /** * Return the Airy function @f$ Ai(x) @f$ of @c float argument x. */ inline float airy_aif(float __x) { float __Ai, __Bi, __Aip, __Bip; std::__detail::__airy<float>(__x, __Ai, __Bi, __Aip, __Bip); return __Ai; } /** * Return the Airy function @f$ Ai(x) @f$ of <tt>long double</tt> argument x. */ inline long double airy_ail(long double __x) { long double __Ai, __Bi, __Aip, __Bip; std::__detail::__airy<long double>(__x, __Ai, __Bi, __Aip, __Bip); return __Ai; } /** * Return the Airy function @f$ Ai(x) @f$ of real argument x. */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type airy_ai(_Tp __x) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; __type __Ai, __Bi, __Aip, __Bip; std::__detail::__airy<__type>(__x, __Ai, __Bi, __Aip, __Bip); return __Ai; } /** * Return the Airy function @f$ Bi(x) @f$ of @c float argument x. */ inline float airy_bif(float __x) { float __Ai, __Bi, __Aip, __Bip; std::__detail::__airy<float>(__x, __Ai, __Bi, __Aip, __Bip); return __Bi; } /** * Return the Airy function @f$ Bi(x) @f$ of <tt>long double</tt> argument x. */ inline long double airy_bil(long double __x) { long double __Ai, __Bi, __Aip, __Bip; std::__detail::__airy<long double>(__x, __Ai, __Bi, __Aip, __Bip); return __Bi; } /** * Return the Airy function @f$ Bi(x) @f$ of real argument x. */ template<typename _Tp> inline typename __gnu_cxx::__promote<_Tp>::__type airy_bi(_Tp __x) { typedef typename __gnu_cxx::__promote<_Tp>::__type __type; __type __Ai, __Bi, __Aip, __Bip; std::__detail::__airy<__type>(__x, __Ai, __Bi, __Aip, __Bip); return __Bi; } // Confluent hypergeometric functions /** * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ * of @c float numeratorial parameter @c a, denominatorial parameter @c c, * and argument @c x. * * @see conf_hyperg for details. */ inline float conf_hypergf(float __a, float __c, float __x) { return std::__detail::__conf_hyperg<float>(__a, __c, __x); } /** * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ * of <tt>long double</tt> numeratorial parameter @c a, * denominatorial parameter @c c, and argument @c x. * * @see conf_hyperg for details. */ inline long double conf_hypergl(long double __a, long double __c, long double __x) { return std::__detail::__conf_hyperg<long double>(__a, __c, __x); } /** * Return the confluent hypergeometric function @f$ {}_1F_1(a;c;x) @f$ * of real numeratorial parameter @c a, denominatorial parameter @c c, * and argument @c x. * * The confluent hypergeometric function is defined by * @f[ * {}_1F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n x^n}{(c)_n n!} * @f] * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$, * @f$ (x)_0 = 1 @f$ * * @param __a The numeratorial parameter * @param __c The denominatorial parameter * @param __x The argument */ template<typename _Tpa, typename _Tpc, typename _Tp> inline typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type conf_hyperg(_Tpa __a, _Tpc __c, _Tp __x) { typedef typename __gnu_cxx::__promote_3<_Tpa, _Tpc, _Tp>::__type __type; return std::__detail::__conf_hyperg<__type>(__a, __c, __x); } // Hypergeometric functions /** * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ * of @ float numeratorial parameters @c a and @c b, * denominatorial parameter @c c, and argument @c x. * * @see hyperg for details. */ inline float hypergf(float __a, float __b, float __c, float __x) { return std::__detail::__hyperg<float>(__a, __b, __c, __x); } /** * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ * of <tt>long double</tt> numeratorial parameters @c a and @c b, * denominatorial parameter @c c, and argument @c x. * * @see hyperg for details. */ inline long double hypergl(long double __a, long double __b, long double __c, long double __x) { return std::__detail::__hyperg<long double>(__a, __b, __c, __x); } /** * Return the hypergeometric function @f$ {}_2F_1(a,b;c;x) @f$ * of real numeratorial parameters @c a and @c b, * denominatorial parameter @c c, and argument @c x. * * The hypergeometric function is defined by * @f[ * {}_2F_1(a;c;x) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n x^n}{(c)_n n!} * @f] * where the Pochhammer symbol is @f$ (x)_k = (x)(x+1)...(x+k-1) @f$, * @f$ (x)_0 = 1 @f$ * * @param __a The first numeratorial parameter * @param __b The second numeratorial parameter * @param __c The denominatorial parameter * @param __x The argument */ template<typename _Tpa, typename _Tpb, typename _Tpc, typename _Tp> inline typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp>::__type hyperg(_Tpa __a, _Tpb __b, _Tpc __c, _Tp __x) { typedef typename __gnu_cxx::__promote_4<_Tpa, _Tpb, _Tpc, _Tp> ::__type __type; return std::__detail::__hyperg<__type>(__a, __b, __c, __x); } // @} _GLIBCXX_END_NAMESPACE_VERSION } // namespace __gnu_cxx #endif // __STRICT_ANSI__ #pragma GCC visibility pop #endif // _GLIBCXX_BITS_SPECFUN_H