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

   1
   2 [section:sph_bessel Spherical Bessel Functions of the First and Second Kinds]
   3
   4 [h4 Synopsis]
   5
   6 `#include <boost/math/special_functions/bessel.hpp>`
   7
   8    template <class T1, class T2>
   9    ``__sf_result`` sph_bessel(unsigned v, T2 x);
  10
  11    template <class T1, class T2, class ``__Policy``>
  12    ``__sf_result`` sph_bessel(unsigned v, T2 x, const ``__Policy``&);
  13
  14    template <class T1, class T2>
  15    ``__sf_result`` sph_neumann(unsigned v, T2 x);
  16
  17    template <class T1, class T2, class ``__Policy``>
  18    ``__sf_result`` sph_neumann(unsigned v, T2 x, const ``__Policy``&);
  19
  20 [h4 Description]
  21
  22 The functions __sph_bessel and __sph_neumann return the result of the
  23 Spherical Bessel functions of the first and second kinds respectively:
  24
  25 sph_bessel(v, x) = j[sub v](x)
  26
  27 sph_neumann(v, x) = y[sub v](x) = n[sub v](x)
  28
  29 where:
  30
  31 [equation sbessel2]
  32
  33 The return type of these functions is computed using the __arg_promotion_rules
  34 for the single argument type T.
  35
  36 [optional_policy]
  37
  38 The functions return the result of __domain_error whenever the result is
  39 undefined or complex: this occurs when `x < 0`.
  40
  41 The j[sub v][space] function is cyclic like J[sub v][space] but differs
  42 in its behaviour at the origin:
  43
  44 [graph sph_bessel]
  45
  46 Likewise y[sub v][space] is also cyclic for large x, but tends to -[infin][space]
  47 for small /x/:
  48
  49 [graph sph_neumann]
  50
  51 [h4 Testing]
  52
  53 There are two sets of test values: spot values calculated using
  54 [@http://functions.wolfram.com/ functions.wolfram.com],
  55 and a much larger set of tests computed using
  56 a simplified version of this implementation
  57 (with all the special case handling removed).
  58
  59 [h4 Accuracy]
  60
  61 [table_sph_bessel]
  62
  63 [table_sph_neumann]
  64
  65 [h4 Implementation]
  66
  67 Other than error handling and a couple of special cases these functions
  68 are implemented directly in terms of their definitions:
  69
  70 [equation sbessel2]
  71
  72 The special cases occur for:
  73
  74 j[sub 0][space]= __sinc_pi(x) = sin(x) / x
  75
  76 and for small ['x < 1], we can use the series:
  77
  78 [equation sbessel5]
  79
  80 which neatly avoids the problem of calculating 0/0 that can occur with the
  81 main definition as x [rarr] 0.
  82
  83 [endsect]
  84
  85 [/
  86   Copyright 2006 John Maddock, Paul A. Bristow and Xiaogang Zhang.
  87   Distributed under the Boost Software License, Version 1.0.
  88   (See accompanying file LICENSE_1_0.txt or copy at
  89   http://www.boost.org/LICENSE_1_0.txt).
  90 ]