Analytic Integral and its applications

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K.Aomoto studied integral representation of special functions and obtained the following results :1) He established an integral formulas for quasi hypergeometric functions and obtained monodromy formulas, and gave an explicit formula of singular point by using Picard-Lefschetz transform.2) He extended the notion of LR transform and density matrices into multi dimensional case. More explicitly, he defined Gram-Schmidt orthogonal polynomials with a given density and he dfined its LR transform.M.Miyake studied partial differential equatons in complex domain and obtained the following results :3) He characterized the Borel summability of divergent formal solution of the Cauchy problem of certain non-Kowalevski type equations. He also gave an integral representation of the Borel sum.4) He characterized a notion of singular equation or singular point for a nonlinear partial differential equation which depends on each solution. Moreover, the characterization of the singulaity is given by showing the convergent or the divergent criterion of the formal solutions.