Computing complex Airy functions by numerical quadrature

Integral representations are considered of solutions of the Airydifferential equation <em>w</em>''-<em>z</em>, <em>w</em>=0 for computing Airy functions for complex values of <em>z</em>.In a first method contour integral representations of the Airyfunctions are written as non-oscillating integrals for obtainingstable representations, which are evaluated by the trapezoidalrule.In a second method an integral representation is evaluated byusing generalized Gauss-Laguerre quadrature; this approachprovides a fast method for computing Airy functions to apredetermined accuracy.Comparisons are made with well-known algorithms of Amos, designedfor computing Bessel functions of complex argument. Severaldiscrepancies with Amos' code are detected, and it is pointed outfor which regions of the complex plane Amos' code is less accurate than thequadrature algorithms.Hints are given in order to build reliable software for complexAiry functions.