arXiv math-ph/05010622005-01-25math-phmath.MP

与Airy函数有关的函数和积分的渐近表示

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

A. I. Nikishov, V. I. Ritus

备注：: 23 pages, 7 figurs

摘要：

得到了函数${\rm Ai}_1(x), {\rm Gi}(x), {\rm Ai}'(x), {\rm Ai}^2(x), {\rm Bi}^ 2(x)$的渐近表示。作为一个副产品，阶乘恒等式（21 '）被发现.导出的渐近表示的积分$\int_v^{\infty}dx{\rm Ai}(x)h(x,v)$的$v\to-\infty$和积分，不同于它的变化${\rm Ai}(x)$由${\rm Ai}'(x)$或${\rm Ai_1(x)}$，提出.以Airy函数${\rm Ai}(z)$为例，将Stokes现象看作是各道次上最陡下降线的不连续行为的结果。当$z$穿过斯托克斯射线时，在较高通路上的最陡下降线突然将其渐近接近在较低通路上的最陡下降线的方向改变为接近该线的相反端的方向。因此，当沿着最陡下降线绘制的积分等值线越过较高的通径时，当$z$穿过斯托克斯射线时，它开始或停止越过较低的通径，结果隐性级数（来自较低通径的贡献）在包含显性级数的函数的渐近表示中不连续地出现或消失。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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