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Vanishing Theorems in Hyperasymptotic Analysis and their Applications

超渐近分析中的消失定理及其应用

基本信息

批准号：
15540158
负责人：
MAJIMA Hideyuki
金额：
$ 2.3万
依托单位：
Ochanomizu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

The vanishing theorem of commutative case in asymptotic analysis means by Cech cohomology that, if asymptotically flat functions are given on the intersections of a good covering of sectors at the infinity in the extended complex plane, then the functions are written in the form of difference of 2 asymptotically developable functions associated the covering. This theorem can be reformed to the case where the functions are with precise estimates. For example, we, Adri B. Olde Daalhuis and the author, can extend to the case of hyperasymptotics of level 1 and 2.It is applied to study the structure of formal solutions to inhomogenous linear differential equations by using solutions asymptotic to the series 0 of the associated homogeneous linear differential equations : inhomogeneous equations associated with the Bessel equation, for example, the Anger function, etc..Related to the study on the Scorrer function, the author established a method of calculating formal solutions of the Airy equation.In the course of the study, the author invited Prof. Werner Balser, specialist of asymptotic analysis, and Prof. Reinhart Schaefke, who were working on asymptotic analysis of several variables, and we, Schaefke and the author, discussed on the extension of vanishing theorems in asymptotic analysis in several variable with a condition which the author had not supposed in his Lecture note in Math 1075, Springer verlag.The author and Obayashi succeeded in calculateing solutions, asymptotic expansions and Stokes multipliers near the singularities of system of partial differential equations of 2 variables, analogous to the Bessel equation or the Kummer equation.

渐近分析中对易情形的消失定理用切赫上同调的方法得到：如果在扩张复平面上无穷远的扇区的交点上给出渐近平坦函数，则该函数以与该覆盖相关的两个渐近可展函数的差的形式表示。这一定理可以改进为函数具有精确估计的情况。例如，我们，Adri B.Olde Daalhuis和作者，可以推广到水平1和2的超渐近性的情况。它被用来研究非齐次线性微分方程列0的渐近解的结构：与Bessel方程有关的非齐次方程，例如，愤怒函数等。在研究Scorrer函数的过程中，作者建立了计算Aary方程的形式解的方法。在研究过程中，作者邀请了渐近分析专家Werner Balser教授和Reinhart Schaefke教授，Schaefke和作者讨论了多元渐近分析中渐近定理的推广，并讨论了作者在《数学1075，Springer Verlages》的讲稿中没有假设的条件下多元渐近分析中的零点定理的推广。作者和大林成功地计算了二元偏微分方程组的奇点附近的解、渐近展开式和Stokes乘子，类似于Bessel方程或Kummer方程。