Computing complex Airy functions by numerical quadrature (Q1604688)

Airy functions are solutions of the differential equation \[ \frac{d^{2}w}{dz^{2}}-zw=0. \] Two linearly independent solutions that are real for real values of \(z\) are denoted by \(\text{Ai}(z)\) and \(\text{Bi}(z)\). They have the integral representation \[ \begin{aligned} {\text{Ai}}(z) &=\frac{1}{\pi}\int_{0}^{\infty}\cos(zt+\frac{t^3}{3}) dt,\\ \text{Bi} (z) &=\frac{1}{\pi}\int_{0}^{\infty}\sin(zt+\frac{t^3}{3}) dt+\frac{1}{\pi}\int_{0}^{\infty}e^{zt-t^3/3} dt, \end{aligned} \] where we assume that \(z\) is real. In this paper the authors are concerned with the numerical evaluation of \(\text{Ai}(z)\) and \(\text{Ai}'(z)\) for complex values of \(z\) by numerical quadrature. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss-Laguerre quadrature. This approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.