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 12The [@http://mathworld.wolfram.com/SincFunction.html Sinus Cardinal family of functions] 13(indexed by the family of indices [^a > 0]) 14is defined by 15 16[equation special_functions_blurb20] 17 18it sees heavy use in signal processing tasks. 19 20By analogy, the 21[@http://mathworld.wolfram.com/SinhcFunction.htm Hyperbolic Sinus Cardinal] 22family of functions 23(also indexed by the family of indices [^a > 0]) is defined by 24 25[equation special_functions_blurb22] 26 27These two families of functions are composed of entire functions. 28 29These functions (__sinc_pi and __sinhc_pi) are needed by 30[@http://www.boost.org/libs/math/quaternion/quaternion.html our implementation] 31of [@http://mathworld.wolfram.com/Quaternion.html quaternions] 32and [@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] [/section:sinc_overview Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview] 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 57Computes 58[link math_toolkit.sinc.sinc_overview 59the Sinus Cardinal] of x: 60 61[expression sinc_pi(x) = sin(x) / x] 62 63The second form is for complex numbers, 64quaternions, octonions etc. Taylor series are used at the origin 65to ensure accuracy. 66 67[graph sinc_pi] 68 69[optional_policy] 70 71[endsect] [/section sinc_pi] 72 73 74[section sinhc_pi] 75 76`` 77#include <boost/math/special_functions/sinhc.hpp> 78`` 79 80 template<class T> 81 ``__sf_result`` sinhc_pi(const T x); 82 83 template<class T, class ``__Policy``> 84 ``__sf_result`` sinhc_pi(const T x, const ``__Policy``&); 85 86 template<typename T, template<typename> class U> 87 U<T> sinhc_pi(const U<T> x); 88 89 template<class T, template<typename> class U, class ``__Policy``> 90 U<T> sinhc_pi(const U<T> x, const ``__Policy``&); 91 92Computes [@http://mathworld.wolfram.com/SinhcFunction.html sinhc function], 93the [link math_toolkit.sinc.sinc_overview Hyperbolic Sinus Cardinal] of x: 94 95[expression sinhc_pi(x) = sinh(x) / x] 96 97The second form is for 98complex numbers, quaternions, octonions etc. Taylor series are used at the origin 99to ensure accuracy. 100 101The return type of the first form is computed using the __arg_promotion_rules 102when T is an integer type. 103 104[optional_policy] 105 106[graph sinhc_pi] 107 108[endsect] [/section sinhc_pi] 109 110[endsect] [/section:sinc Sinus Cardinal and Hyperbolic Sinus Cardinal Functions] 111 112
