报告人：Gergő Nemes

题目: Dingle's final main rule, Berry's transition, and Howls' conjecture 

地点：理学楼401

时间：2024年8月6日， 9:00~10:00

内容：The Stokes phenomenon refers to 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 later proposed a set of rules for locating Stokes lines and continuing asymptotic expansions across them. One of these rules, the final main rule, states that half the discontinuity in form occurs upon reaching the Stokes line, and half upon leaving it.

M. V. Berry demonstrated that if an asymptotic expansion is terminated just before its numerically smallest term, the transition between two different asymptotic forms across a Stokes line happens smoothly, not discontinuously, as traditionally interpreted. On a Stokes line, Berry's law predicts a multiplier of 1/2 for the emerging small exponentials, in accordance with Dingle's final main rule.

In this talk, we examine two closely related asymptotic expansions where the multipliers of exponentially small contributions may deviate from Dingle's rule: their values can differ from 1/2 on a Stokes line and be non-zero only on the line itself. This unusual behaviour of the multipliers results from a series 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 by C. J. Howls. 

报告人简介：报告人在匈牙利罗兰大学获得理学学士和理学硕士学位，研究渐近分析和特殊函数。首次明确提出了伽马函数的斯特林渐近展开式系数的公式。在中欧大学攻读博士学位期间，专注于拉普拉斯型积分的复苏问题，为特殊函数提供了可计算的误差界限，并在Mathematische Annalen、Proceedings of the Royal Society A等期刊上发表。随后在爱丁堡大学和阿尔弗雷德·雷尼数学研究所任职。已发表论文37篇，引用次数超过330次。曾获得包括Simons基金会、日本学术振兴会和匈牙利科学院等机构的嘉奖，获得多项奖项和资助。报告人还为NIST数学函数数字图书馆做出了贡献，组织会议，并指导员工。