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

   1
   2 [section:hankel Hankel Functions]
   3 [section:cyl_hankel Cyclic Hankel Functions]
   4
   5 [h4 Synopsis]
   6
   7    template <class T1, class T2>
   8    std::complex<``__sf_result``> cyl_hankel_1(T1 v, T2 x);
   9
  10    template <class T1, class T2, class ``__Policy``>
  11    std::complex<``__sf_result``> cyl_hankel_1(T1 v, T2 x, const ``__Policy``&);
  12
  13    template <class T1, class T2>
  14    std::complex<``__sf_result``> cyl_hankel_2(T1 v, T2 x);
  15
  16    template <class T1, class T2, class ``__Policy``>
  17    std::complex<``__sf_result``> cyl_hankel_2(T1 v, T2 x, const ``__Policy``&);
  18
  19
  20 [h4 Description]
  21
  22 The functions __cyl_hankel_1 and __cyl_hankel_2 return the result of the
  23 [@http://dlmf.nist.gov/10.2#P3 Hankel functions] of the first and second kind respectively:
  24
  25 [:['cyl_hankel_1(v, x) = H[sub v][super (1)](x) = J[sub v](x) + i Y[sub v](x)]]
  26
  27 [:['cyl_hankel_2(v, x) = H[sub v][super (2)](x) = J[sub v](x) - i Y[sub v](x)]]
  28
  29 where:
  30
  31 ['J[sub v](x)] is the Bessel function of the first kind, and ['Y[sub v](x)] is the Bessel function of the second kind.
  32
  33 The return type of these functions is computed using the __arg_promotion_rules
  34 when T1 and T2 are different types.  The functions are also optimised for the
  35 relatively common case that T1 is an integer.
  36
  37 [optional_policy]
  38
  39 Note that while the arguments to these functions are real values, the results are complex.
  40 That means that the functions can only be instantiated on types `float`, `double` and `long double`.
  41 The functions have also been extended to operate over the whole range of ['v] and ['x]
  42 (unlike __cyl_bessel_j and __cyl_neumann).
  43
  44 [h4 Performance]
  45
  46 These functions are generally more efficient than two separate calls to the underlying Bessel
  47 functions as internally Bessel J and Y can be computed simultaneously.
  48
  49 [h4 Testing]
  50
  51 There are just a few spot tests to exercise all the special case handling - the bulk of the testing is done
  52 on the Bessel functions upon which these are based.
  53
  54 [h4 Accuracy]
  55
  56 Refer to __cyl_bessel_j and __cyl_neumann.
  57
  58 [h4 Implementation]
  59
  60 For ['x < 0] the following reflection formulae are used:
  61
  62 [@http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/16/01/01/ [equation hankel1]]
  63
  64 [@http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/16/01/01/ [equation hankel2]]
  65
  66 [@http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/16/01/01/ [equation hankel3]]
  67
  68 Otherwise the implementation is trivially in terms of the Bessel J and Y functions.
  69
  70 Note however, that the Hankel functions compute the Bessel J and Y functions simultaneously,
  71 and therefore a single Hankel function call is more efficient than two Bessel function calls.
  72 The one exception is when ['v] is a small positive integer, in which case the usual Bessel function
  73 routines for integer order are used.
  74
  75 [endsect]
  76
  77
  78 [section:sph_hankel Spherical Hankel Functions]
  79
  80 [h4 Synopsis]
  81
  82    template <class T1, class T2>
  83    std::complex<``__sf_result``> sph_hankel_1(T1 v, T2 x);
  84
  85    template <class T1, class T2, class ``__Policy``>
  86    std::complex<``__sf_result``> sph_hankel_1(T1 v, T2 x, const ``__Policy``&);
  87
  88    template <class T1, class T2>
  89    std::complex<``__sf_result``> sph_hankel_2(T1 v, T2 x);
  90
  91    template <class T1, class T2, class ``__Policy``>
  92    std::complex<``__sf_result``> sph_hankel_2(T1 v, T2 x, const ``__Policy``&);
  93
  94
  95 [h4 Description]
  96
  97 The functions __sph_hankel_1 and __sph_hankel_2 return the result of the
  98 [@http://dlmf.nist.gov/10.47#P1 spherical Hankel functions] of the first and second kind respectively:
  99
 100 [equation hankel4]
 101
 102 [equation hankel5]
 103
 104 The return type of these functions is computed using the __arg_promotion_rules
 105 when T1 and T2 are different types.  The functions are also optimised for the
 106 relatively common case that T1 is an integer.
 107
 108 [optional_policy]
 109
 110 Note that while the arguments to these functions are real values, the results are complex.
 111 That means that the functions can only be instantiated on types `float`, `double` and `long double`.
 112 The functions have also been extended to operate over the whole range of ['v] and ['x]
 113 (unlike __cyl_bessel_j and __cyl_neumann).
 114
 115 [h4 Testing]
 116
 117 There are just a few spot tests to exercise all the special case handling - the bulk of the testing is done
 118 on the Bessel functions upon which these are based.
 119
 120 [h4 Accuracy]
 121
 122 Refer to __cyl_bessel_j and __cyl_neumann.
 123
 124 [h4 Implementation]
 125
 126 These functions are trivially implemented in terms of __cyl_hankel_1 and __cyl_hankel_2.
 127
 128 [endsect]
 129 [endsect]
 130
 131 [/
 132   Copyright 2012 John Maddock.
 133   Distributed under the Boost Software License, Version 1.0.
 134   (See accompanying file LICENSE_1_0.txt or copy at
 135   http://www.boost.org/LICENSE_1_0.txt).
 136 ]