On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions

TL;DR

Problem: computing inhomogeneous Airy functions $Gi(z)$ and $Hi(z)$ for general complex $z$ is challenged by oscillatory integral representations. Approach: the authors derive stable non-oscillating integral representations by applying steepest-descent deformation to the standard integrals and by sector-aware reduction using connection formulas, enabling direct quadrature in many regions. Contributions: explicit integral formulas for $Hi(z)$ and $Gi(z)$ across the complex plane, with practical algorithms and numerical illustrations validating stability and agreement with known asymptotics and Scorer data. Impact: enables robust cross-domain computation of Scorer functions and provides methodology potentially extendable to Airy functions and other special functions.

Abstract

Integral representations are considered of solutions of the inhomogeneous Airy differential equation $w''-z w=\pm1/π$. The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain non-oscillating integrals for complex values of $z$. In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results.