论文标题
丁格尔的最终主要规则，贝瑞的过渡和how叫的猜想
Dingle's final main rule, Berry's transition, and Howls' conjecture
论文作者
论文摘要
Stokes现象是在复杂平面（称为Stokes line）跨某些射线的渐近膨胀形式的明显不连续的变化，因为被指数小的术语预先出现的其他扩展出现在其表示中。 G. G. Stokes首先在研究通风功能的渐近行为时观察到。 R. B. Dingle提出了一组用于定位Stokes线路并持续渐近扩展的规则。在这些规则中包括“最终主要规则”，指出形式的一半是在到达Stokes Line时发生的一半，而将其留在另一侧时。 M. V. Berry证明，如果渐近扩张在数值最小的术语之前终止，则跨stokes线之间的两种不同的渐近形式之间的过渡会顺利进行，而不是不连续的，就像在Stokes现象的常规解释中一样。在Stokes Line上，根据Dingle的最终主要规则，Berry的定律预测了新兴小指数的乘以$ \ frac {1} {2} $的乘数。在本文中，我们考虑了两个密切相关的渐近扩展，其中指数较小的贡献的乘数可能不再遵守丁格尔的规则：它们的价值与Stokes Line上的$ \ frac {1} {2} $有所不同，并且只能在线本身上是非零。乘数的这种异常行为是一系列高阶Stokes现象的结果。我们表明，这些现象是快速的，但在一系列最佳截短的高空抑制重新膨胀的剩余条件下是快速的，但平稳的转变。为此，我们验证了C. J. Howls引起的猜想。
The Stokes phenomenon is the apparent discontinuous change in the form of the asymptotic expansion of a function across certain rays in the complex plane, known as Stokes lines, as additional expansions, pre-factored by exponentially small terms, appear in its representation. It was first observed by G. G. Stokes while studying the asymptotic behaviour of the Airy function. R. B. Dingle proposed a set of rules for locating Stokes lines and continuing asymptotic expansions across them. Included among these rules is the "final main rule" stating that half the discontinuity in form occurs on reaching the Stokes line, and half on leaving it the other side. M. V. Berry demonstrated that, if an asymptotic expansion is terminated just before its numerically least term, the transition between two different asymptotic forms across a Stokes line is effected smoothly and not discontinuously as in the conventional interpretation of the Stokes phenomenon. On a Stokes line, in accordance with Dingle's final main rule, Berry's law predicts a multiplier of $\frac{1}{2}$ for the emerging small exponentials. In this paper, we consider two closely related asymptotic expansions in which the multipliers of exponentially small contributions may no longer obey Dingle's rule: their values can differ from $\frac{1}{2}$ on a Stokes line and can be non-zero only on the line itself. This unusual behaviour of the multipliers is a result of a sequence of higher-order Stokes phenomena. We show that these phenomena are rapid but smooth transitions in the remainder terms of a series of optimally truncated hyperasymptotic re-expansions. To this end, we verify a conjecture due to C. J. Howls.
