Eigenfunctions of the Airyʼs integral transform: Properties, numerical computations and asymptotic behaviors

This paper is devoted to the study of the spectrum of the integral operator with Airyʼs kernel. We provide the reader with some efficient methods of computing the eigenfunctions and the eigenvalues of this operator. The first method is based on the use of a differential operator which commutes with this integral operator. The second method is based on a Gaussian quadrature method applied to the finite Airyʼs transform. The asymptotic behavior of the eigenfunctions is studied by the use of a WKB method. Some numerical examples are given to illustrate results of this work.