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