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Zbl 1012.33001
Gil, Amparo; Segura, Javier; Temme, Nico M.
Computing complex Airy functions by numerical quadrature. (English)
[J] Numer. Algorithms 30, No.1, 11-23 (2002). ISSN 1017-1398; ISSN 1572-9265/e

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 $$\align {\text{Ai}}(z) &=\frac{1}{\pi}\int_{0}^{\infty}\cos(zt+\frac{t^3}{3}) dt,\\ \text{Bi} (z) &=\frac{1}{\pi}\int_{0}^{\infty}\sin(zt+\frac{t^3}{3}) dt+\frac{1}{\pi}\int_{0}^{\infty}e^{zt-t^3/3} dt, \endalign$$ 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.
[Stamatis Koumandos (Nicosia)]

MSC 2000:: *33C10 Cylinder functions, etc.
33F05 Numerical approximation of special functions
41A60 Asymptotic problems in approximation
30E10 Approximation in the complex domain
65D20 Computation of special functions
65D32 Quadrature formulas (numerical methods)

Keywords: Airy functions; numerical computation of special functions; numerical quadrature

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