[/
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]

[section:airy_root Finding Zeros of Airy Functions]

[h4 Synopsis]

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

Functions for obtaining both a single zero or root of the Airy functions,
and placing multiple zeros into a container like `std::vector`
by providing an output iterator.

The signature of the single value functions are:

  template <class T>
  T airy_ai_zero(
           int m);         // 1-based index of zero.

  template <class T>
  T airy_bi_zero(
           int m);         // 1-based index of zero.

and for multiple zeros:

 template <class T, class OutputIterator>
 OutputIterator airy_ai_zero(
                      int start_index,           // 1-based index of first zero.
                      unsigned number_of_zeros,  // How many zeros to generate.
                      OutputIterator out_it);    // Destination for zeros.

 template <class T, class OutputIterator>
 OutputIterator airy_bi_zero(
                      int start_index,           // 1-based index of zero.
                      unsigned number_of_zeros,  // How many zeros to generate
                      OutputIterator out_it);    // Destination for zeros.

There are also versions which allow control of the __policy_section for error handling and precision.

  template <class T>
  T airy_ai_zero(
           int m,          // 1-based index of zero.
           const Policy&); // Policy to use.

  template <class T>
  T airy_bi_zero(
           int m,          // 1-based index of zero.
           const Policy&); // Policy to use.


 template <class T, class OutputIterator>
 OutputIterator airy_ai_zero(
                      int start_index,           // 1-based index of first zero.
                      unsigned number_of_zeros,  // How many zeros to generate.
                      OutputIterator out_it,     // Destination for zeros.
                      const Policy& pol);        // Policy to use.

 template <class T, class OutputIterator>
 OutputIterator airy_bi_zero(
                      int start_index,           // 1-based index of zero.
                      unsigned number_of_zeros,  // How many zeros to generate.
                      OutputIterator out_it,     // Destination for zeros.
                      const Policy& pol);        // Policy to use.

[h4 Description]

The Airy Ai and Bi functions have an infinite
number of zeros on the negative real axis. The real zeros on the negative real
axis can be found by solving for the roots of

[emquad] ['Ai(x[sub m]) = 0]

[emquad] ['Bi(y[sub m]) = 0]

Here, ['x[sub m]] represents the ['m[super th]]
root of the Airy Ai function,
and ['y[sub m]] represents the ['m[super th]]
root of the Airy Bi function.

The zeros or roots (values of `x` where the function crosses the horizontal `y = 0` axis)
of the Airy Ai and Bi functions are computed by two functions,
`airy_ai_zero` and `airy_bi_zero`.

In each case the index or rank of the zero
returned is 1-based, which is to say:

   airy_ai_zero(1);

returns the first zero of Ai.

Passing an `start_index <= 0` results in a __domain_error being raised.

The first few zeros returned by these functions have approximate values as follows:

[table
[[m][Ai][Bi]]
[[1][-2.33811...][-1.17371...]]
[[2][-4.08795...][-3.27109...]]
[[3][-5.52056...][-4.83074...]]
[[4][-6.78671...][-6.16985...]]
[[5][-7.94413...][-7.37676...]]
[[6][-9.02265...][-8.49195...]]
]

[graph airy_zeros]


[h4 Examples of finding Airy Zeros]

[import ../../example/airy_zeros_example.cpp]

[airy_zeros_example_1]
[airy_zeros_example_2]

Produces the program output:
[pre
boost::math::airy_ai_zero<double>(1) = -2.33811
boost::math::airy_ai_zero<double>(2) = -4.08795
boost::math::airy_bi_zero<double>(3) = -4.83074
airy_ai_zeros:
-2.33811
-4.08795
-5.52056
-6.78671
-7.94413

boost::math::airy_bi_zero<float_type>(1)  = -2.3381074104597670384891972524467354406385401456711
boost::math::airy_bi_zero<float_type>(2)  = -4.0879494441309706166369887014573910602247646991085
boost::math::airy_bi_zero<float_type>(7)  = -9.5381943793462388866329885451560196208390720763825
airy_ai_zeros:
-2.3381074104597670384891972524467354406385401456711
-4.0879494441309706166369887014573910602247646991085
-5.5205598280955510591298555129312935737972142806175
]

The full code (and output) for this example is at
[@../../example/airy_zeros_example.cpp airy_zeros_example.cpp],

[h3 Implementation]

Given the following function (A&S 10.4.105):

[equation airy_zero_1]

Then an initial estimate for the n[super th] zero a[sub n] of Ai is given by (A&S 10.4.94):

[equation airy_zero_2]

and an initial estimate for the n[super th] zero b[sub n] of Bi is given by (A&S 10.4.98):

[equation airy_zero_3]

Thereafter the roots are refined using Newton iteration.


[h3 Testing]

The precision of evaluation of zeros was tested at 50 decimal digits using `cpp_dec_float_50`
and found identical with spot values computed by __WolframAlpha.

[endsect]  [/section:bessel Finding Zeros of Bessel Functions of the First and Second Kinds]


[endsect]

