[/
Copyright (c) 2012 John Maddock
Use, modification and distribution are subject to 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)
]

[section:airy Airy Functions]

[section:ai Airy Ai Function]

[heading Synopsis]

``
  #include <boost/math/special_functions/airy.hpp>
``

  namespace boost { namespace math {

   template <class T>
   ``__sf_result`` airy_ai(T x);

   template <class T, class Policy>
   ``__sf_result`` airy_ai(T x, const Policy&);

  }} // namespaces
  
[heading Description]

The function __airy_ai calculates the Airy function Ai which is the first solution
to the differential equation:

[equation airy]

See Weisstein, Eric W. "Airy Functions." From MathWorld--A Wolfram Web Resource.
[@http://mathworld.wolfram.com/AiryFunctions.html];


[optional_policy]

The following graph illustrates how this function changes as /x/ changes: for negative /x/
the function is cyclic, while for positive /x/ the value tends to zero:

[graph airy_ai]

[heading Accuracy]

This function is implemented entirely in terms of the Bessel functions 
__cyl_bessel_j and __cyl_bessel_k - refer to those functions for detailed accuracy information.

In general though, the relative error is low (less than 100 [epsilon]) for /x > 0/ while
only the absolute error is low for /x < 0/.

[heading Testing]

Since this function is implemented in terms of other special functions, there are only a few 
basic sanity checks, using test values from [@http://functions.wolfram.com/ Wolfram Airy Functions].

[heading Implementation]

This function is implemented in terms of the Bessel functions using the relations:

[equation airy_ai]

[endsect]

[section:bi Airy Bi Function]

[heading Synopsis]

``
  #include <boost/math/special_functions/airy.hpp>
``

  namespace boost { namespace math {

   template <class T>
   ``__sf_result`` airy_bi(T x);

   template <class T, class Policy>
   ``__sf_result`` airy_bi(T x, const Policy&);

  }} // namespaces
  
[heading Description]

The function __airy_bi calculates the Airy function Bi which is the second solution to the differential equation:

[equation airy]

[optional_policy]

The following graph illustrates how this function changes as /x/ changes: for negative /x/
the function is cyclic, while for positive /x/ the value tends to infinity:

[graph airy_bi]

[heading Accuracy]

This function is implemented entirely in terms of the Bessel functions 
__cyl_bessel_i and __cyl_bessel_j - refer to those functions for detailed accuracy information.

In general though, the relative error is low (less than 100 [epsilon]) for /x > 0/ while
only the absolute error is low for /x < 0/.

[heading Testing]

Since this function is implemented in terms of other special functions, there are only a few 
basic sanity checks, using test values from [@http://functions.wolfram.com functions.wolfram.com].

[heading Implementation]

This function is implemented in terms of the Bessel functions using the relations:

[equation airy_bi]

[endsect]

[section:aip Airy Ai' Function]

[heading Synopsis]

``
  #include <boost/math/special_functions/airy.hpp>
``

  namespace boost { namespace math {

   template <class T>
   ``__sf_result`` airy_ai_prime(T x);

   template <class T, class Policy>
   ``__sf_result`` airy_ai_prime(T x, const Policy&);

  }} // namespaces
  
[heading Description]

The function __airy_ai_prime calculates the Airy function Ai' which is the derivative of the first solution to the differential equation:

[equation airy]

[optional_policy]

The following graph illustrates how this function changes as /x/ changes: for negative /x/
the function is cyclic, while for positive /x/ the value tends to zero:

[graph airy_aip]

[heading Accuracy]

This function is implemented entirely in terms of the Bessel functions 
__cyl_bessel_j and __cyl_bessel_k - refer to those functions for detailed accuracy information.

In general though, the relative error is low (less than 100 [epsilon]) for /x > 0/ while
only the absolute error is low for /x < 0/.

[heading Testing]

Since this function is implemented in terms of other special functions, there are only a few 
basic sanity checks, using test values from [@http://functions.wolfram.com functions.wolfram.com].

[heading Implementation]

This function is implemented in terms of the Bessel functions using the relations:

[equation airy_aip]

[endsect]

[section:bip Airy Bi' Function]

[heading Synopsis]

``
  #include <boost/math/special_functions/airy.hpp>
``

  namespace boost { namespace math {

   template <class T>
   ``__sf_result`` airy_bi_prime(T x);

   template <class T, class Policy>
   ``__sf_result`` airy_bi_prime(T x, const Policy&);

  }} // namespaces
  
[heading Description]

The function __airy_bi_prime calculates the Airy function Bi' which is the derivative of the second solution to the differential equation:

[equation airy]

[optional_policy]

The following graph illustrates how this function changes as /x/ changes: for negative /x/
the function is cyclic, while for positive /x/ the value tends to infinity:

[graph airy_bi]

[heading Accuracy]

This function is implemented entirely in terms of the Bessel functions 
__cyl_bessel_i and __cyl_bessel_j - refer to those functions for detailed accuracy information.

In general though, the relative error is low (less than 100 [epsilon]) for /x > 0/ while
only the absolute error is low for /x < 0/.

[heading Testing]

Since this function is implemented in terms of other special functions, there are only a few 
basic sanity checks, using test values from [@http://functions.wolfram.com functions.wolfram.com].

[heading Implementation]

This function is implemented in terms of the Bessel functions using the relations:

[equation airy_bip]

[endsect]

[endsect]