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

   1 [/ math.qbk
   2   Copyright 2006 Hubert Holin and John Maddock.
   3   Distributed under the Boost Software License, Version 1.0.
   4   (See accompanying file LICENSE_1_0.txt or copy at
   5   http://www.boost.org/LICENSE_1_0.txt).
   6 ]
   7
   8 [section:sinc Sinus Cardinal and Hyperbolic Sinus Cardinal Functions]
   9
  10 [section:sinc_overview Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview]
  11
  12 The [@http://mathworld.wolfram.com/SincFunction.html Sinus Cardinal family of functions]
  13 (indexed by the family of indices [^a > 0])
  14 is defined by
  15
  16 [equation special_functions_blurb20]
  17
  18 it sees heavy use in signal processing tasks.
  19
  20 By analogy, the
  21 [@http://mathworld.wolfram.com/SinhcFunction.htm Hyperbolic Sinus Cardinal]
  22 family of functions
  23 (also indexed by the family of indices [^a > 0]) is defined by
  24
  25 [equation special_functions_blurb22]
  26
  27 These two families of functions are composed of entire functions.
  28
  29 These functions (__sinc_pi and __sinhc_pi) are needed by
  30 [@http://www.boost.org/libs/math/quaternion/quaternion.html our implementation]
  31 of [@http://mathworld.wolfram.com/Quaternion.html quaternions]
  32 and [@http://mathworld.wolfram.com/Octonion.html octonions].
  33
  34 [: ['[*Sinus Cardinal of index pi (purple) and Hyperbolic Sinus Cardinal of index pi (red) on R]]]
  35 [: [$../graphs/sinc_pi_and_sinhc_pi_on_r.png]]
  36
  37 [endsect]
  38
  39 [section sinc_pi]
  40
  41 ``
  42 #include <boost/math/special_functions/sinc.hpp>
  43 ``
  44
  45    template<class T>
  46    ``__sf_result`` sinc_pi(const T x);
  47
  48    template<class T, class ``__Policy``>
  49    ``__sf_result`` sinc_pi(const T x, const ``__Policy``&);
  50
  51    template<class T, template<typename> class U>
  52    U<T> sinc_pi(const U<T> x);
  53
  54    template<class T, template<typename> class U, class ``__Policy``>
  55    U<T> sinc_pi(const U<T> x, const ``__Policy``&);
  56
  57 Computes
  58 [link math_toolkit.sinc.sinc_overview
  59 the Sinus Cardinal] of x:
  60
  61    sinc_pi(x) = sin(x) / x
  62
  63 The second form is for complex numbers,
  64 quaternions, octonions etc. Taylor series are used at the origin
  65 to ensure accuracy.
  66
  67 [graph sinc_pi]
  68
  69 [optional_policy]
  70
  71 [endsect]
  72
  73 [section sinhc_pi]
  74
  75 ``
  76 #include <boost/math/special_functions/sinhc.hpp>
  77 ``
  78
  79    template<class T>
  80    ``__sf_result`` sinhc_pi(const T x);
  81
  82    template<class T, class ``__Policy``>
  83    ``__sf_result`` sinhc_pi(const T x, const ``__Policy``&);
  84
  85    template<typename T, template<typename> class U>
  86    U<T> sinhc_pi(const U<T> x);
  87
  88    template<class T, template<typename> class U, class ``__Policy``>
  89    U<T> sinhc_pi(const U<T> x, const ``__Policy``&);
  90
  91 Computes http://mathworld.wolfram.com/SinhcFunction.html
  92 [link math_toolkit.sinc.sinc_overview
  93 the Hyperbolic Sinus Cardinal] of x:
  94
  95    sinhc_pi(x) = sinh(x) / x
  96
  97 The second form is for
  98 complex numbers, quaternions, octonions etc. Taylor series are used at the origin
  99 to ensure accuracy.
 100
 101 The return type of the first form is computed using the __arg_promotion_rules
 102 when T is an integer type.
 103
 104 [optional_policy]
 105
 106 [graph sinhc_pi]
 107
 108 [endsect]
 109
 110 [endsect]