special_functions/detail/hypergeometric_1F1_large_a.hpp

0001
0002 ///////////////////////////////////////////////////////////////////////////////
0003 //  Copyright 2018 John Maddock
0004 //  Distributed under the Boost
0005 //  Software License, Version 1.0. (See accompanying file
0006 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
0007 //
0008 #ifndef BOOST_MATH_HYPERGEOMETRIC_1F1_CF_HPP
0009 #define BOOST_MATH_HYPERGEOMETRIC_1F1_CF_HPP
0010 //
0011 // Evaluation of 1F1 by continued fraction
0012 // by asymptotic approximation for large a,
0013 // see https://dlmf.nist.gov/13.8#E9
0014 //
0015 // This is not terribly useful, as it only gets a few digits correct even for very
0016 // large a, also needs b and z small:
0017 //
0018
0019
0020   namespace boost { namespace math { namespace detail {
0021
0022      template <class T, class Policy>
0023      T hypergeometric_1F1_large_neg_a_asymtotic_dlmf_13_8_9(T a, T b, T z, const Policy& pol)
0024      {
0025         T result = boost::math::cyl_bessel_j(b - 1, sqrt(2 * z * (b - 2 * a)), pol);
0026         result *= boost::math::tgamma(b, pol) * exp(z / 2);
0027         T p = pow((b / 2 - a) * z, (1 - b) / 4);
0028         result *= p;
0029         result *= p;
0030         return result;
0031      }
0032
0033   } } } // namespaces
0034
0035 #endif // BOOST_MATH_HYPERGEOMETRIC_1F1_BESSEL_HPP