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Airy Ai Function

#include <boost/math/special_functions/airy.hpp>
namespace boost { namespace math {

 template <class T>
 calculated-result-type airy_ai(T x);

 template <class T, class Policy>
 calculated-result-type airy_ai(T x, const Policy&);

}} // namespaces

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

See Weisstein, Eric W. "Airy Functions." From MathWorld--A Wolfram Web Resource.;

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

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:


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 ε) for x > 0 while only the absolute error is low for x < 0 as the following error plot illustrates:


Since this function is implemented in terms of other special functions, there are only a few basic sanity checks, using test values from Wolfram Airy Functions.


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