# Boost C++ Libraries

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### boost/math/special_functions/detail/bessel_jn.hpp

```//  Copyright (c) 2006 Xiaogang Zhang
//  Use, modification and distribution are subject to the
//  Boost Software License, Version 1.0. (See accompanying file

#ifndef BOOST_MATH_BESSEL_JN_HPP
#define BOOST_MATH_BESSEL_JN_HPP

#ifdef _MSC_VER
#pragma once
#endif

#include <boost/math/special_functions/detail/bessel_j0.hpp>
#include <boost/math/special_functions/detail/bessel_j1.hpp>
#include <boost/math/special_functions/detail/bessel_jy.hpp>
#include <boost/math/special_functions/detail/bessel_jy_asym.hpp>
#include <boost/math/special_functions/detail/bessel_jy_series.hpp>

// Bessel function of the first kind of integer order
// J_n(z) is the minimal solution
// n < abs(z), forward recurrence stable and usable
// n >= abs(z), forward recurrence unstable, use Miller's algorithm

namespace boost { namespace math { namespace detail{

template <typename T, typename Policy>
T bessel_jn(int n, T x, const Policy& pol)
{
T value(0), factor, current, prev, next;

BOOST_MATH_STD_USING

//
// Reflection has to come first:
//
if (n < 0)
{
factor = static_cast<T>((n & 0x1) ? -1 : 1);  // J_{-n}(z) = (-1)^n J_n(z)
n = -n;
}
else
{
factor = 1;
}
if(x < 0)
{
factor *= (n & 0x1) ? -1 : 1;  // J_{n}(-z) = (-1)^n J_n(z)
x = -x;
}
//
// Special cases:
//
if(asymptotic_bessel_large_x_limit(T(n), x))
return factor * asymptotic_bessel_j_large_x_2<T>(T(n), x, pol);
if (n == 0)
{
return factor * bessel_j0(x);
}
if (n == 1)
{
return factor * bessel_j1(x);
}

if (x == 0)                             // n >= 2
{
return static_cast<T>(0);
}

BOOST_MATH_ASSERT(n > 1);
T scale = 1;
if (n < abs(x))                         // forward recurrence
{
prev = bessel_j0(x);
current = bessel_j1(x);
policies::check_series_iterations<T>("boost::math::bessel_j_n<%1%>(%1%,%1%)", n, pol);
for (int k = 1; k < n; k++)
{
T fact = 2 * k / x;
//
// rescale if we would overflow or underflow:
//
if((fabs(fact) > 1) && ((tools::max_value<T>() - fabs(prev)) / fabs(fact) < fabs(current)))
{
scale /= current;
prev /= current;
current = 1;
}
value = fact * current - prev;
prev = current;
current = value;
}
}
else if((x < 1) || (n > x * x / 4) || (x < 5))
{
return factor * bessel_j_small_z_series(T(n), x, pol);
}
else                                    // backward recurrence
{
T fn; int s;                        // fn = J_(n+1) / J_n
// |x| <= n, fast convergence for continued fraction CF1
boost::math::detail::CF1_jy(static_cast<T>(n), x, &fn, &s, pol);
prev = fn;
current = 1;
// Check recursion won't go on too far:
policies::check_series_iterations<T>("boost::math::bessel_j_n<%1%>(%1%,%1%)", n, pol);
for (int k = n; k > 0; k--)
{
T fact = 2 * k / x;
if((fabs(fact) > 1) && ((tools::max_value<T>() - fabs(prev)) / fabs(fact) < fabs(current)))
{
prev /= current;
scale /= current;
current = 1;
}
next = fact * current - prev;
prev = current;
current = next;
}
value = bessel_j0(x) / current;       // normalization
scale = 1 / scale;
}
value *= factor;

if(tools::max_value<T>() * scale < fabs(value))
return policies::raise_overflow_error<T>("boost::math::bessel_jn<%1%>(%1%,%1%)", nullptr, pol);

return value / scale;
}

}}} // namespaces

#endif // BOOST_MATH_BESSEL_JN_HPP

```