Computing complex Airy functions by numerical quadrature
type: Item
instance of: scholarly article
title: Computing complex Airy functions by numerical quadrature
zbMATH Open document ID: 1012.33001
DOI: 10.1023/A:1015636825525
author: Amparo Gil
author: Javier Segura
author: Nico M. Temme
published in: Numerical Algorithms
publication date: 2002-07-08T00:00:00Z
review text: 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.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 33F05
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 65D20
Mathematics Subject Classification ID: 65D32
zbMATH DE Number: 1764747
zbMATH Keywords: Airy functions
zbMATH Keywords: numerical computation of special functions
zbMATH Keywords: numerical quadrature
MaRDI profile type: Publication
full work available at URL: a:1015636825525
OpenAlex ID: W2467351019
description: scientific article; zbMATH DE number 1764747