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111 lines
2.9 KiB
Plaintext
111 lines
2.9 KiB
Plaintext
[/ math.qbk
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Copyright 2006 Hubert Holin and John Maddock.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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[section:sinc Sinus Cardinal and Hyperbolic Sinus Cardinal Functions]
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[section:sinc_overview Sinus Cardinal and Hyperbolic Sinus Cardinal Functions Overview]
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The [@http://mathworld.wolfram.com/SincFunction.html Sinus Cardinal family of functions]
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(indexed by the family of indices [^a > 0])
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is defined by
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[equation special_functions_blurb20]
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it sees heavy use in signal processing tasks.
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By analogy, the
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[@http://mathworld.wolfram.com/SinhcFunction.htm Hyperbolic Sinus Cardinal]
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family of functions
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(also indexed by the family of indices [^a > 0]) is defined by
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[equation special_functions_blurb22]
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These two families of functions are composed of entire functions.
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These functions (__sinc_pi and __sinhc_pi) are needed by
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[@http://www.boost.org/libs/math/quaternion/quaternion.html our implementation]
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of [@http://mathworld.wolfram.com/Quaternion.html quaternions]
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and [@http://mathworld.wolfram.com/Octonion.html octonions].
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[: ['[*Sinus Cardinal of index pi (purple) and Hyperbolic Sinus Cardinal of index pi (red) on R]]]
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[: [$../graphs/sinc_pi_and_sinhc_pi_on_r.png]]
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[endsect]
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[section sinc_pi]
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``
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#include <boost/math/special_functions/sinc.hpp>
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``
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template<class T>
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``__sf_result`` sinc_pi(const T x);
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template<class T, class ``__Policy``>
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``__sf_result`` sinc_pi(const T x, const ``__Policy``&);
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template<class T, template<typename> class U>
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U<T> sinc_pi(const U<T> x);
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template<class T, template<typename> class U, class ``__Policy``>
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U<T> sinc_pi(const U<T> x, const ``__Policy``&);
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Computes
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[link math_toolkit.sinc.sinc_overview
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the Sinus Cardinal] of x:
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sinc_pi(x) = sin(x) / x
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The second form is for complex numbers,
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quaternions, octonions etc. Taylor series are used at the origin
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to ensure accuracy.
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[graph sinc_pi]
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[optional_policy]
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[endsect]
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[section sinhc_pi]
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``
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#include <boost/math/special_functions/sinhc.hpp>
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``
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template<class T>
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``__sf_result`` sinhc_pi(const T x);
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template<class T, class ``__Policy``>
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``__sf_result`` sinhc_pi(const T x, const ``__Policy``&);
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template<typename T, template<typename> class U>
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U<T> sinhc_pi(const U<T> x);
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template<class T, template<typename> class U, class ``__Policy``>
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U<T> sinhc_pi(const U<T> x, const ``__Policy``&);
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Computes http://mathworld.wolfram.com/SinhcFunction.html
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[link math_toolkit.sinc.sinc_overview
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the Hyperbolic Sinus Cardinal] of x:
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sinhc_pi(x) = sinh(x) / x
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The second form is for
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complex numbers, quaternions, octonions etc. Taylor series are used at the origin
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to ensure accuracy.
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The return type of the first form is computed using the __arg_promotion_rules
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when T is an integer type.
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[optional_policy]
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[graph sinhc_pi]
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[endsect]
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[endsect]
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