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110 lines
2.9 KiB
Plaintext
110 lines
2.9 KiB
Plaintext
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[section:bessel_derivatives Derivatives of the Bessel Functions]
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[h4 Synopsis]
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`#include <boost/math/special_functions/bessel_prime.hpp>`
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template <class T1, class T2>
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``__sf_result`` cyl_bessel_j_prime(T1 v, T2 x);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` cyl_bessel_j_prime(T1 v, T2 x, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` cyl_neumann_prime(T1 v, T2 x);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` cyl_neumann_prime(T1 v, T2 x, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` cyl_bessel_i_prime(T1 v, T2 x);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` cyl_bessel_i_prime(T1 v, T2 x, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` cyl_bessel_k_prime(T1 v, T2 x);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` cyl_bessel_k_prime(T1 v, T2 x, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` sph_bessel_prime(T1 v, T2 x);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` sph_bessel_prime(T1 v, T2 x, const ``__Policy``&);
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template <class T1, class T2>
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``__sf_result`` sph_neumann_prime(T1 v, T2 x);
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template <class T1, class T2, class ``__Policy``>
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``__sf_result`` sph_neumann_prime(T1 v, T2 x, const ``__Policy``&);
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[h4 Description]
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These functions return the first derivative with respect to /x/ of the corresponding Bessel function.
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The return type of these functions is computed using the __arg_promotion_rules
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when T1 and T2 are different types. The functions are also optimised for the
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relatively common case that T1 is an integer.
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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.
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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://www.wolframalpha.com/ wolframalpha.com],
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and a much larger set of tests computed using
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a relation to the underlying Bessel functions that the implementation
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does not use.
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[h4 Accuracy]
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The accuracy of these functions is broadly similar to the underlying Bessel functions.
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[table_cyl_bessel_i_prime_integer_orders_]
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[table_cyl_bessel_i_prime]
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[table_cyl_bessel_j_prime_integer_orders_]
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[table_cyl_bessel_j_prime]
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[table_cyl_bessel_k_prime_integer_orders_]
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[table_cyl_bessel_k_prime]
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[table_sph_bessel_prime]
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[table_sph_neumann_prime]
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[h4 Implementation]
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In the general case, the derivatives are calculated using the relations:
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[equation bessel_derivatives1]
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There are also a number of special cases, for large x we have:
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[equation bessel_derivatives4]
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And for small x:
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[equation bessel_derivatives5]
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[endsect]
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[/
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Copyright 2013, 2013 John Maddock, Anton Bikineev.
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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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