
[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:

[:['cyl_hankel_1(v, x) = H[sub v][super (1)](x) = J[sub v](x) + i Y[sub v](x)]]

[:['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: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]
[endsect]

[/ 
  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).
]
