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- [section:hankel Hankel Functions]
- [section:cyl_hankel Cyclic Hankel Functions]
- [h4 Synopsis]
- template <class T1, class T2>
- std::complex<``__sf_result``> cyl_hankel_1(T1 v, T2 x);
- template <class T1, class T2, class ``__Policy``>
- std::complex<``__sf_result``> cyl_hankel_1(T1 v, T2 x, const ``__Policy``&);
- template <class T1, class T2>
- std::complex<``__sf_result``> cyl_hankel_2(T1 v, T2 x);
-
- template <class T1, class T2, class ``__Policy``>
- std::complex<``__sf_result``> cyl_hankel_2(T1 v, T2 x, const ``__Policy``&);
-
-
- [h4 Description]
- The functions __cyl_hankel_1 and __cyl_hankel_2 return the result of the
- [@http://dlmf.nist.gov/10.2#P3 Hankel functions] of the first and second kind respectively:
- [expression ['cyl_hankel_1(v, x) = H[sub v][super (1)](x) = J[sub v](x) + i Y[sub v](x)]]
- [expression ['cyl_hankel_2(v, x) = H[sub v][super (2)](x) = J[sub v](x) - i Y[sub v](x)]]
- where:
- ['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.
- The return type of these functions is computed using the __arg_promotion_rules
- when T1 and T2 are different types. The functions are also optimised for the
- relatively common case that T1 is an integer.
- [optional_policy]
- Note that while the arguments to these functions are real values, the results are complex.
- That means that the functions can only be instantiated on types `float`, `double` and `long double`.
- The functions have also been extended to operate over the whole range of ['v] and ['x]
- (unlike __cyl_bessel_j and __cyl_neumann).
- [h4 Performance]
- These functions are generally more efficient than two separate calls to the underlying Bessel
- functions as internally Bessel J and Y can be computed simultaneously.
- [h4 Testing]
- There are just a few spot tests to exercise all the special case handling - the bulk of the testing is done
- on the Bessel functions upon which these are based.
- [h4 Accuracy]
- Refer to __cyl_bessel_j and __cyl_neumann.
- [h4 Implementation]
- For ['x < 0] the following reflection formulae are used:
- [@http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/16/01/01/ [equation hankel1]]
- [@http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/16/01/01/ [equation hankel2]]
- [@http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/16/01/01/ [equation hankel3]]
- Otherwise the implementation is trivially in terms of the Bessel J and Y functions.
- Note however, that the Hankel functions compute the Bessel J and Y functions simultaneously,
- and therefore a single Hankel function call is more efficient than two Bessel function calls.
- The one exception is when ['v] is a small positive integer, in which case the usual Bessel function
- routines for integer order are used.
- [endsect] [/section:cyl_hankel Cyclic Hankel Functions]
- [section:sph_hankel Spherical Hankel Functions]
- [h4 Synopsis]
- template <class T1, class T2>
- std::complex<``__sf_result``> sph_hankel_1(T1 v, T2 x);
- template <class T1, class T2, class ``__Policy``>
- std::complex<``__sf_result``> sph_hankel_1(T1 v, T2 x, const ``__Policy``&);
- template <class T1, class T2>
- std::complex<``__sf_result``> sph_hankel_2(T1 v, T2 x);
-
- template <class T1, class T2, class ``__Policy``>
- std::complex<``__sf_result``> sph_hankel_2(T1 v, T2 x, const ``__Policy``&);
-
-
- [h4 Description]
- The functions __sph_hankel_1 and __sph_hankel_2 return the result of the
- [@http://dlmf.nist.gov/10.47#P1 spherical Hankel functions] of the first and second kind respectively:
- [equation hankel4]
- [equation hankel5]
- The return type of these functions is computed using the __arg_promotion_rules
- when T1 and T2 are different types. The functions are also optimised for the
- relatively common case that T1 is an integer.
- [optional_policy]
- Note that while the arguments to these functions are real values, the results are complex.
- That means that the functions can only be instantiated on types `float`, `double` and `long double`.
- The functions have also been extended to operate over the whole range of ['v] and ['x]
- (unlike __cyl_bessel_j and __cyl_neumann).
- [h4 Testing]
- There are just a few spot tests to exercise all the special case handling - the bulk of the testing is done
- on the Bessel functions upon which these are based.
- [h4 Accuracy]
- Refer to __cyl_bessel_j and __cyl_neumann.
- [h4 Implementation]
- These functions are trivially implemented in terms of __cyl_hankel_1 and __cyl_hankel_2.
- [endsect] [/section:sph_hankel Spherical Hankel Functions]
- [endsect] [/section:hankel Hankel Functions]
- [/
- Copyright 2012 John Maddock.
- Distributed under the Boost Software License, Version 1.0.
- (See accompanying file LICENSE_1_0.txt or copy at
- http://www.boost.org/LICENSE_1_0.txt).
- ]
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