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Computing complex Airy functions by numerical quadrature. (English) Zbl 1012.33001

Airy functions are solutions of the differential equation

$\frac{{d}^{2}w}{d{z}^{2}}-zw=0·$

Two linearly independent solutions that are real for real values of $z$ are denoted by $\text{Ai}\left(z\right)$ and $\text{Bi}\left(z\right)$. They have the integral representation

$\begin{array}{cc}\hfill \text{Ai}\left(z\right)& =\frac{1}{\pi }{\int }_{0}^{\infty }cos\left(zt+\frac{{t}^{3}}{3}\right)dt,\hfill \\ \hfill \text{Bi}\left(z\right)& =\frac{1}{\pi }{\int }_{0}^{\infty }sin\left(zt+\frac{{t}^{3}}{3}\right)dt+\frac{1}{\pi }{\int }_{0}^{\infty }{e}^{zt-{t}^{3}/3}dt,\hfill \end{array}$

where we assume that $z$ is real. In this paper the authors are concerned with the numerical evaluation of $\text{Ai}\left(z\right)$ and ${\text{Ai}}^{\text{'}}\left(z\right)$ 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.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 33F05 Numerical approximation and evaluation of special functions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 30E10 Approximation in the complex domain 65D20 Computation of special functions, construction of tables 65D32 Quadrature and cubature formulas (numerical methods)