1391 lines
46 KiB
C++
1391 lines
46 KiB
C++
// Mathematical Special Functions for -*- C++ -*-
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// Copyright (C) 2006-2022 Free Software Foundation, Inc.
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//
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// This file is part of the GNU ISO C++ Library. This library is free
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// software; you can redistribute it and/or modify it under the
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// terms of the GNU General Public License as published by the
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// Free Software Foundation; either version 3, or (at your option)
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// any later version.
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// This library is distributed in the hope that it will be useful,
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// but WITHOUT ANY WARRANTY; without even the implied warranty of
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// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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// GNU General Public License for more details.
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// Under Section 7 of GPL version 3, you are granted additional
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// permissions described in the GCC Runtime Library Exception, version
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// 3.1, as published by the Free Software Foundation.
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// You should have received a copy of the GNU General Public License and
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// a copy of the GCC Runtime Library Exception along with this program;
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// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
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// <http://www.gnu.org/licenses/>.
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/** @file bits/specfun.h
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* This is an internal header file, included by other library headers.
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* Do not attempt to use it directly. @headername{cmath}
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*/
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#ifndef _GLIBCXX_BITS_SPECFUN_H
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#define _GLIBCXX_BITS_SPECFUN_H 1
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#pragma GCC visibility push(default)
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#include <bits/c++config.h>
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#define __STDCPP_MATH_SPEC_FUNCS__ 201003L
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#define __cpp_lib_math_special_functions 201603L
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#if __cplusplus <= 201403L && __STDCPP_WANT_MATH_SPEC_FUNCS__ == 0
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# error include <cmath> and define __STDCPP_WANT_MATH_SPEC_FUNCS__
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#endif
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#include <bits/stl_algobase.h>
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#include <limits>
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#include <type_traits>
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#include <tr1/gamma.tcc>
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#include <tr1/bessel_function.tcc>
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#include <tr1/beta_function.tcc>
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#include <tr1/ell_integral.tcc>
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#include <tr1/exp_integral.tcc>
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#include <tr1/hypergeometric.tcc>
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#include <tr1/legendre_function.tcc>
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#include <tr1/modified_bessel_func.tcc>
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#include <tr1/poly_hermite.tcc>
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#include <tr1/poly_laguerre.tcc>
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#include <tr1/riemann_zeta.tcc>
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namespace std _GLIBCXX_VISIBILITY(default)
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{
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_GLIBCXX_BEGIN_NAMESPACE_VERSION
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/**
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* @defgroup mathsf Mathematical Special Functions
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* @ingroup numerics
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*
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* @section mathsf_desc Mathematical Special Functions
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*
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* A collection of advanced mathematical special functions,
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* defined by ISO/IEC IS 29124 and then added to ISO C++ 2017.
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*
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*
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* @subsection mathsf_intro Introduction and History
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* The first significant library upgrade on the road to C++2011,
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* <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2005/n1836.pdf">
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* TR1</a>, included a set of 23 mathematical functions that significantly
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* extended the standard transcendental functions inherited from C and declared
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* in @<cmath@>.
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*
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* Although most components from TR1 were eventually adopted for C++11 these
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* math functions were left behind out of concern for implementability.
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* The math functions were published as a separate international standard
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* <a href="http://www.open-std.org/JTC1/SC22/WG21/docs/papers/2010/n3060.pdf">
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* IS 29124 - Extensions to the C++ Library to Support Mathematical Special
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* Functions</a>.
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*
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* For C++17 these functions were incorporated into the main standard.
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*
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* @subsection mathsf_contents Contents
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* The following functions are implemented in namespace @c std:
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* - @ref assoc_laguerre "assoc_laguerre - Associated Laguerre functions"
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* - @ref assoc_legendre "assoc_legendre - Associated Legendre functions"
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* - @ref beta "beta - Beta functions"
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* - @ref comp_ellint_1 "comp_ellint_1 - Complete elliptic functions of the first kind"
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* - @ref comp_ellint_2 "comp_ellint_2 - Complete elliptic functions of the second kind"
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* - @ref comp_ellint_3 "comp_ellint_3 - Complete elliptic functions of the third kind"
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* - @ref cyl_bessel_i "cyl_bessel_i - Regular modified cylindrical Bessel functions"
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* - @ref cyl_bessel_j "cyl_bessel_j - Cylindrical Bessel functions of the first kind"
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* - @ref cyl_bessel_k "cyl_bessel_k - Irregular modified cylindrical Bessel functions"
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* - @ref cyl_neumann "cyl_neumann - Cylindrical Neumann functions or Cylindrical Bessel functions of the second kind"
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* - @ref ellint_1 "ellint_1 - Incomplete elliptic functions of the first kind"
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* - @ref ellint_2 "ellint_2 - Incomplete elliptic functions of the second kind"
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* - @ref ellint_3 "ellint_3 - Incomplete elliptic functions of the third kind"
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* - @ref expint "expint - The exponential integral"
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* - @ref hermite "hermite - Hermite polynomials"
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* - @ref laguerre "laguerre - Laguerre functions"
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* - @ref legendre "legendre - Legendre polynomials"
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* - @ref riemann_zeta "riemann_zeta - The Riemann zeta function"
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* - @ref sph_bessel "sph_bessel - Spherical Bessel functions"
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* - @ref sph_legendre "sph_legendre - Spherical Legendre functions"
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* - @ref sph_neumann "sph_neumann - Spherical Neumann functions"
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*
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* The hypergeometric functions were stricken from the TR29124 and C++17
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* versions of this math library because of implementation concerns.
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* However, since they were in the TR1 version and since they are popular
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* we kept them as an extension in namespace @c __gnu_cxx:
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* - @ref __gnu_cxx::conf_hyperg "conf_hyperg - Confluent hypergeometric functions"
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* - @ref __gnu_cxx::hyperg "hyperg - Hypergeometric functions"
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*
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* <!-- @subsection mathsf_general General Features -->
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*
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* @subsection mathsf_promotion Argument Promotion
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* The arguments suppled to the non-suffixed functions will be promoted
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* according to the following rules:
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* 1. If any argument intended to be floating point is given an integral value
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* That integral value is promoted to double.
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* 2. All floating point arguments are promoted up to the largest floating
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* point precision among them.
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*
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* @subsection mathsf_NaN NaN Arguments
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* If any of the floating point arguments supplied to these functions is
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* invalid or NaN (std::numeric_limits<Tp>::quiet_NaN),
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* the value NaN is returned.
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*
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* @subsection mathsf_impl Implementation
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*
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* We strive to implement the underlying math with type generic algorithms
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* to the greatest extent possible. In practice, the functions are thin
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* wrappers that dispatch to function templates. Type dependence is
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* controlled with std::numeric_limits and functions thereof.
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*
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* We don't promote @c float to @c double or @c double to <tt>long double</tt>
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* reflexively. The goal is for @c float functions to operate more quickly,
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* at the cost of @c float accuracy and possibly a smaller domain of validity.
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* Similaryly, <tt>long double</tt> should give you more dynamic range
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* and slightly more pecision than @c double on many systems.
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*
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* @subsection mathsf_testing Testing
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*
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* These functions have been tested against equivalent implementations
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* from the <a href="http://www.gnu.org/software/gsl">
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* Gnu Scientific Library, GSL</a> and
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* <a href="http://www.boost.org/doc/libs/1_60_0/libs/math/doc/html/index.html">Boost</a>
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* and the ratio
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* @f[
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* \frac{|f - f_{test}|}{|f_{test}|}
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* @f]
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* is generally found to be within 10<sup>-15</sup> for 64-bit double on
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* linux-x86_64 systems over most of the ranges of validity.
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*
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* @todo Provide accuracy comparisons on a per-function basis for a small
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* number of targets.
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*
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* @subsection mathsf_bibliography General Bibliography
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*
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* @see Abramowitz and Stegun: Handbook of Mathematical Functions,
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* with Formulas, Graphs, and Mathematical Tables
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* Edited by Milton Abramowitz and Irene A. Stegun,
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* National Bureau of Standards Applied Mathematics Series - 55
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* Issued June 1964, Tenth Printing, December 1972, with corrections
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* Electronic versions of A&S abound including both pdf and navigable html.
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* @see for example http://people.math.sfu.ca/~cbm/aands/
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*
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* @see The old A&S has been redone as the
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* NIST Digital Library of Mathematical Functions: http://dlmf.nist.gov/
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* This version is far more navigable and includes more recent work.
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*
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* @see An Atlas of Functions: with Equator, the Atlas Function Calculator
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* 2nd Edition, by Oldham, Keith B., Myland, Jan, Spanier, Jerome
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*
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* @see Asymptotics and Special Functions by Frank W. J. Olver,
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* Academic Press, 1974
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*
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* @see Numerical Recipes in C, The Art of Scientific Computing,
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* by William H. Press, Second Ed., Saul A. Teukolsky,
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* William T. Vetterling, and Brian P. Flannery,
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* Cambridge University Press, 1992
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*
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* @see The Special Functions and Their Approximations: Volumes 1 and 2,
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* by Yudell L. Luke, Academic Press, 1969
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*
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* @{
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*/
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// Associated Laguerre polynomials
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/**
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* Return the associated Laguerre polynomial of order @c n,
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* degree @c m: @f$ L_n^m(x) @f$ for @c float argument.
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*
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* @see assoc_laguerre for more details.
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*/
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inline float
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assoc_laguerref(unsigned int __n, unsigned int __m, float __x)
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{ return __detail::__assoc_laguerre<float>(__n, __m, __x); }
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/**
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* Return the associated Laguerre polynomial of order @c n,
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* degree @c m: @f$ L_n^m(x) @f$.
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*
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* @see assoc_laguerre for more details.
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*/
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inline long double
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assoc_laguerrel(unsigned int __n, unsigned int __m, long double __x)
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{ return __detail::__assoc_laguerre<long double>(__n, __m, __x); }
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/**
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* Return the associated Laguerre polynomial of nonnegative order @c n,
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* nonnegative degree @c m and real argument @c x: @f$ L_n^m(x) @f$.
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*
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* The associated Laguerre function of real degree @f$ \alpha @f$,
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* @f$ L_n^\alpha(x) @f$, is defined by
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* @f[
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* L_n^\alpha(x) = \frac{(\alpha + 1)_n}{n!}
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* {}_1F_1(-n; \alpha + 1; x)
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* @f]
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* where @f$ (\alpha)_n @f$ is the Pochhammer symbol and
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* @f$ {}_1F_1(a; c; x) @f$ is the confluent hypergeometric function.
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*
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* The associated Laguerre polynomial is defined for integral
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* degree @f$ \alpha = m @f$ by:
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* @f[
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* L_n^m(x) = (-1)^m \frac{d^m}{dx^m} L_{n + m}(x)
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* @f]
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* where the Laguerre polynomial is defined by:
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* @f[
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* L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (x^ne^{-x})
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* @f]
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* and @f$ x >= 0 @f$.
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* @see laguerre for details of the Laguerre function of degree @c n
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*
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* @tparam _Tp The floating-point type of the argument @c __x.
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* @param __n The order of the Laguerre function, <tt>__n >= 0</tt>.
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* @param __m The degree of the Laguerre function, <tt>__m >= 0</tt>.
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* @param __x The argument of the Laguerre function, <tt>__x >= 0</tt>.
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* @throw std::domain_error if <tt>__x < 0</tt>.
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*/
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template<typename _Tp>
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inline typename __gnu_cxx::__promote<_Tp>::__type
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assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x)
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{
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typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
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return __detail::__assoc_laguerre<__type>(__n, __m, __x);
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}
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// Associated Legendre functions
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/**
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* Return the associated Legendre function of degree @c l and order @c m
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* for @c float argument.
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*
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* @see assoc_legendre for more details.
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*/
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inline float
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assoc_legendref(unsigned int __l, unsigned int __m, float __x)
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{ return __detail::__assoc_legendre_p<float>(__l, __m, __x); }
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/**
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* Return the associated Legendre function of degree @c l and order @c m.
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*
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* @see assoc_legendre for more details.
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*/
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inline long double
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assoc_legendrel(unsigned int __l, unsigned int __m, long double __x)
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{ return __detail::__assoc_legendre_p<long double>(__l, __m, __x); }
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/**
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* Return the associated Legendre function of degree @c l and order @c m.
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*
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* The associated Legendre function is derived from the Legendre function
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* @f$ P_l(x) @f$ by the Rodrigues formula:
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* @f[
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* P_l^m(x) = (1 - x^2)^{m/2}\frac{d^m}{dx^m}P_l(x)
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* @f]
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* @see legendre for details of the Legendre function of degree @c l
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*
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* @tparam _Tp The floating-point type of the argument @c __x.
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* @param __l The degree <tt>__l >= 0</tt>.
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* @param __m The order <tt>__m <= l</tt>.
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* @param __x The argument, <tt>abs(__x) <= 1</tt>.
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* @throw std::domain_error if <tt>abs(__x) > 1</tt>.
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*/
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template<typename _Tp>
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inline typename __gnu_cxx::__promote<_Tp>::__type
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assoc_legendre(unsigned int __l, unsigned int __m, _Tp __x)
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{
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typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
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return __detail::__assoc_legendre_p<__type>(__l, __m, __x);
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}
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// Beta functions
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/**
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* Return the beta function, @f$ B(a,b) @f$, for @c float parameters @c a, @c b.
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*
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* @see beta for more details.
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*/
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inline float
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betaf(float __a, float __b)
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{ return __detail::__beta<float>(__a, __b); }
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/**
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* Return the beta function, @f$B(a,b)@f$, for long double
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* parameters @c a, @c b.
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*
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* @see beta for more details.
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*/
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inline long double
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betal(long double __a, long double __b)
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{ return __detail::__beta<long double>(__a, __b); }
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/**
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* Return the beta function, @f$B(a,b)@f$, for real parameters @c a, @c b.
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*
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* The beta function is defined by
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* @f[
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* B(a,b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt
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* = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
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* @f]
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* where @f$ a > 0 @f$ and @f$ b > 0 @f$
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*
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* @tparam _Tpa The floating-point type of the parameter @c __a.
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* @tparam _Tpb The floating-point type of the parameter @c __b.
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* @param __a The first argument of the beta function, <tt> __a > 0 </tt>.
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* @param __b The second argument of the beta function, <tt> __b > 0 </tt>.
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* @throw std::domain_error if <tt> __a < 0 </tt> or <tt> __b < 0 </tt>.
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*/
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template<typename _Tpa, typename _Tpb>
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inline typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type
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beta(_Tpa __a, _Tpb __b)
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{
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typedef typename __gnu_cxx::__promote_2<_Tpa, _Tpb>::__type __type;
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return __detail::__beta<__type>(__a, __b);
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}
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// Complete elliptic integrals of the first kind
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/**
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* Return the complete elliptic integral of the first kind @f$ E(k) @f$
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* for @c float modulus @c k.
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*
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* @see comp_ellint_1 for details.
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*/
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inline float
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comp_ellint_1f(float __k)
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{ return __detail::__comp_ellint_1<float>(__k); }
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/**
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* Return the complete elliptic integral of the first kind @f$ E(k) @f$
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* for long double modulus @c k.
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*
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* @see comp_ellint_1 for details.
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*/
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inline long double
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comp_ellint_1l(long double __k)
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{ return __detail::__comp_ellint_1<long double>(__k); }
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/**
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* Return the complete elliptic integral of the first kind
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* @f$ K(k) @f$ for real modulus @c k.
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*
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* The complete elliptic integral of the first kind is defined as
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* @f[
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* K(k) = F(k,\pi/2) = \int_0^{\pi/2}\frac{d\theta}
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* {\sqrt{1 - k^2 sin^2\theta}}
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* @f]
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* where @f$ F(k,\phi) @f$ is the incomplete elliptic integral of the
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* first kind and the modulus @f$ |k| <= 1 @f$.
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* @see ellint_1 for details of the incomplete elliptic function
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* of the first kind.
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*
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* @tparam _Tp The floating-point type of the modulus @c __k.
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* @param __k The modulus, <tt> abs(__k) <= 1 </tt>
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* @throw std::domain_error if <tt> abs(__k) > 1 </tt>.
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*/
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template<typename _Tp>
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inline typename __gnu_cxx::__promote<_Tp>::__type
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comp_ellint_1(_Tp __k)
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{
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typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
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return __detail::__comp_ellint_1<__type>(__k);
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}
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// Complete elliptic integrals of the second kind
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/**
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* Return the complete elliptic integral of the second kind @f$ E(k) @f$
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* for @c float modulus @c k.
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*
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* @see comp_ellint_2 for details.
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*/
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inline float
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comp_ellint_2f(float __k)
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{ return __detail::__comp_ellint_2<float>(__k); }
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/**
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* Return the complete elliptic integral of the second kind @f$ E(k) @f$
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* for long double modulus @c k.
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*
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* @see comp_ellint_2 for details.
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*/
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inline long double
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comp_ellint_2l(long double __k)
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{ return __detail::__comp_ellint_2<long double>(__k); }
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/**
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* Return the complete elliptic integral of the second kind @f$ E(k) @f$
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* for real modulus @c k.
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*
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* The complete elliptic integral of the second kind is defined as
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* @f[
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* E(k) = E(k,\pi/2) = \int_0^{\pi/2}\sqrt{1 - k^2 sin^2\theta}
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* @f]
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* where @f$ E(k,\phi) @f$ is the incomplete elliptic integral of the
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* second kind and the modulus @f$ |k| <= 1 @f$.
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* @see ellint_2 for details of the incomplete elliptic function
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* of the second kind.
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*
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* @tparam _Tp The floating-point type of the modulus @c __k.
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* @param __k The modulus, @c abs(__k) <= 1
|
|
* @throw std::domain_error if @c abs(__k) > 1.
|
|
*/
|
|
template<typename _Tp>
|
|
inline typename __gnu_cxx::__promote<_Tp>::__type
|
|
comp_ellint_2(_Tp __k)
|
|
{
|
|
typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
|
|
return __detail::__comp_ellint_2<__type>(__k);
|
|
}
|
|
|
|
// Complete elliptic integrals of the third kind
|
|
|
|
/**
|
|
* @brief Return the complete elliptic integral of the third kind
|
|
* @f$ \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.
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|
*
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* @see hyperg for details.
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|
*/
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|
inline long double
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|
hypergl(long double __a, long double __b, long double __c, long double __x)
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|
{ 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
|