On the efficient calculation of the incomplete Airy function with application to edge diffraction

When the geometrical theory of diffraction (GTD) and its extensions are used to calculate fields across transition regions or in focal areas, various special functions are necessary to create uniform representations. One such transition function is the incomplete Airy function, which is applicable when two reflection points are arbitrarily close to each other and the endpoint of the domain considered. An efficient calculation of the incomplete Airy function is presented and shown to equal known asymptotic expansions when its argument is away from the critical point. By using these expansions, uniform field expressions are found which are consistent with the GTD and maintain its computational efficiency.