  Linear second order differential equations having a large real parameter and
turning point in the complex plane are considered. Classical asymptotic
expansions for solutions involve the Airy function and its derivative, along
with two infinite series, the coefficients of which are usually difficult to
compute. By considering the series as asymptotic expansions for two explicitly
defined analytic functions, Cauchy's integral formula is employed to compute
the coefficient functions to high order of accuracy. The method employs a
certain exponential form of Liouville-Green expansions for solutions of the
differential equation, as well as for the Airy function. We illustrate the use
of the method with the high accuracy computation of Airy-type expansions of
Bessel functions of complex argument.
