Asymptotic representations for some functions and integrals connected with the Airy function

Abstract

The asymptotic representations of the functions ${\rm Ai}_1(x), {\rm Gi}(x), {\rm Ai}'(x), {\rm Ai}^2(x), {\rm Bi}^ 2(x)$ are obtained. As a by-product, the factorial identity (21') is found. The derivation of asymptotic representations of the integral $\int_v^{\infty}dx{\rm Ai}(x)h(x,v)$ for $v\to-\infty$ and integrals, differing from it by the change of ${\rm Ai}(x)$ by ${\rm Ai}'(x)$ or ${\rm Ai_1(x)}$, is presented. For the Airy function ${\rm Ai}(z)$, as an example, the Stokes' phenomenon is considered as a consequence of discontinuous behavior of steepest descent lines over the passes. When $z$ crosses the Stokes ray, the steepest descent line over the higher pass abruptly changes the direction of its asymptotic approach to the steepest descent line over the lower pass to the direction of approach to the opposite end of this line. Therefore, when the integration contour, drawn along the steepest descent lines, goes over the higher pass, it begins or stops to go over the lower pass while $z$ crosses the Stokes ray, and as a result the recessive series (contribution from the lower pass) discontinuously appears or disappears in the asymptotic representation of a function containing the dominant series.