// 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