ceph/src/boost/libs/math/doc/sf/bessel_prime.qbk

   1
   2 [section:bessel_derivatives Derivatives of the Bessel Functions]
   3
   4 [h4 Synopsis]
   5
   6 `#include <boost/math/special_functions/bessel_prime.hpp>`
   7
   8    template <class T1, class T2>
   9    ``__sf_result`` cyl_bessel_j_prime(T1 v, T2 x);
  10
  11    template <class T1, class T2, class ``__Policy``>
  12    ``__sf_result`` cyl_bessel_j_prime(T1 v, T2 x, const ``__Policy``&);
  13
  14    template <class T1, class T2>
  15    ``__sf_result`` cyl_neumann_prime(T1 v, T2 x);
  16
  17    template <class T1, class T2, class ``__Policy``>
  18    ``__sf_result`` cyl_neumann_prime(T1 v, T2 x, const ``__Policy``&);
  19
  20    template <class T1, class T2>
  21    ``__sf_result`` cyl_bessel_i_prime(T1 v, T2 x);
  22
  23    template <class T1, class T2, class ``__Policy``>
  24    ``__sf_result`` cyl_bessel_i_prime(T1 v, T2 x, const ``__Policy``&);
  25
  26    template <class T1, class T2>
  27    ``__sf_result`` cyl_bessel_k_prime(T1 v, T2 x);
  28
  29    template <class T1, class T2, class ``__Policy``>
  30    ``__sf_result`` cyl_bessel_k_prime(T1 v, T2 x, const ``__Policy``&);
  31
  32    template <class T1, class T2>
  33    ``__sf_result`` sph_bessel_prime(T1 v, T2 x);
  34
  35    template <class T1, class T2, class ``__Policy``>
  36    ``__sf_result`` sph_bessel_prime(T1 v, T2 x, const ``__Policy``&);
  37
  38    template <class T1, class T2>
  39    ``__sf_result`` sph_neumann_prime(T1 v, T2 x);
  40
  41    template <class T1, class T2, class ``__Policy``>
  42    ``__sf_result`` sph_neumann_prime(T1 v, T2 x, const ``__Policy``&);
  43
  44
  45 [h4 Description]
  46
  47 These functions return the first derivative with respect to /x/ of the corresponding Bessel function.
  48
  49 The return type of these functions is computed using the __arg_promotion_rules
  50 when T1 and T2 are different types.  The functions are also optimised for the
  51 relatively common case that T1 is an integer.
  52
  53 [optional_policy]
  54
  55 The functions return the result of __domain_error whenever the result is
  56 undefined or complex.
  57
  58 [h4 Testing]
  59
  60 There are two sets of test values: spot values calculated using
  61 [@http://www.wolframalpha.com/ wolframalpha.com],
  62 and a much larger set of tests computed using
  63 a relation to the underlying Bessel functions that the implementation
  64 does not use.
  65
  66 [h4 Accuracy]
  67
  68 The accuracy of these functions is broadly similar to the underlying Bessel functions.
  69
  70 [table_cyl_bessel_i_prime_integer_orders_]
  71
  72 [table_cyl_bessel_i_prime]
  73
  74 [table_cyl_bessel_j_prime_integer_orders_]
  75
  76 [table_cyl_bessel_j_prime]
  77
  78 [table_cyl_bessel_k_prime_integer_orders_]
  79
  80 [table_cyl_bessel_k_prime]
  81
  82 [table_sph_bessel_prime]
  83
  84 [table_sph_neumann_prime]
  85
  86
  87 [h4 Implementation]
  88
  89 In the general case, the derivatives are calculated using the relations:
  90
  91 [equation bessel_derivatives1]
  92
  93 There are also a number of special cases, for large x we have:
  94
  95 [equation bessel_derivatives4]
  96
  97 And for small x:
  98
  99 [equation bessel_derivatives5]
 100
 101 [endsect]
 102
 103 [/
 104   Copyright 2013, 2013 John Maddock, Anton Bikineev.
 105
 106   Distributed under the Boost Software License, Version 1.0.
 107   (See accompanying file LICENSE_1_0.txt or copy at
 108   http://www.boost.org/LICENSE_1_0.txt).
 109 ]