Vanishing Theorems in Hyperasymptotic Analysis and their Applications

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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.