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[section:sph_bessel Spherical Bessel Functions of the First and Second Kinds]
[h4 Synopsis]
`#include <boost/math/special_functions/bessel.hpp>`
template <class T1, class T2>
BOOST_MATH_GPU_ENABLED ``__sf_result`` sph_bessel(unsigned v, T2 x);
template <class T1, class T2, class ``__Policy``>
BOOST_MATH_GPU_ENABLED ``__sf_result`` sph_bessel(unsigned v, T2 x, const ``__Policy``&);
template <class T1, class T2>
BOOST_MATH_GPU_ENABLED ``__sf_result`` sph_neumann(unsigned v, T2 x);
template <class T1, class T2, class ``__Policy``>
BOOST_MATH_GPU_ENABLED ``__sf_result`` sph_neumann(unsigned v, T2 x, const ``__Policy``&);
[h4 Description]
The functions __sph_bessel and __sph_neumann return the result of the
Spherical Bessel functions of the first and second kinds respectively:
[:sph_bessel(v, x) = j[sub v](x)]
[:sph_neumann(v, x) = y[sub v](x) = n[sub v](x)]
where:
[equation sbessel2]
The return type of these functions is computed using the __arg_promotion_rules
for the single argument type T.
[optional_policy]
The functions return the result of __domain_error whenever the result is
undefined or complex: this occurs when `x < 0`.
The j[sub v] function is cyclic like J[sub v] but differs in its behaviour at the origin:
[graph sph_bessel]
Likewise y[sub v] is also cyclic for large x, but tends to -[infin]
for small /x/:
[graph sph_neumann]
[h4 Testing]
There are two sets of test values: spot values calculated using
[@http://functions.wolfram.com/ functions.wolfram.com],
and a much larger set of tests computed using
a simplified version of this implementation
(with all the special case handling removed).
[h4 Accuracy]
[table_sph_bessel]
[table_sph_neumann]
[h4 Implementation]
Other than error handling and a couple of special cases these functions
are implemented directly in terms of their definitions:
[equation sbessel2]
The special cases occur for:
[:j[sub 0]= __sinc_pi(x) = sin(x) / x]
and for small ['x < 1], we can use the series:
[equation sbessel5]
which neatly avoids the problem of calculating 0/0 that can occur with the
main definition as x [rarr] 0.
[endsect] [/section:sph_bessel Spherical Bessel Functions of the First and Second Kinds]
[/
Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang.
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or copy at
http://www.boost.org/LICENSE_1_0.txt).
]
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