Algebraic Analysis of Exponential Asymptotics
Project/Area Number  09640234 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Kinki University 
Principal Investigator 
AOKI Takashi Kinki University, Department of Mathematics, Professor, 理工学部, 教授 (80159285)

CoInvestigator(Kenkyūbuntansha) 
OWA Shigeyoshi Kinki University, Dept.of Mathematics, Professor, 理工学部, 教授 (50088506)
IZUMI Shuzo Kinki University, Dept.of Mathematics, Professor, 理工学部, 教授 (80025410)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1998 : ¥900,000 (Direct Cost : ¥900,000)

Keywords  Stokes geometry / Painleve equations / 3d order O.D.E. / transcendence / exponential asymptotics / pvalent functions / Duffing equation / multiplescale analysis / formal functions / ストークス曲線 / パンルヴェ方程式 / 3階線型微分方程式 / 超越性 / 指数漸近級数 / p葉関数 / 多重スケール解析 / ダフィング方程式 / 楕円関数 / Duffing方程式 
Research Abstract 
・2paramnetered exponential asymptotic solution to the Dufling equation is constructed by using the multiplescale analysis. It is proved that the formal solution converges and coincides with the ・Fourier series expansion of the solution written in terms of Jacobi's elliptic function obtained by quadrature. ・The Stokes geometry of the second and the third Painleve equations are determined. ・An Ansatz concerning determination of Stokes gepmetry of third order ordinary differential equations with a large parameter is proposed. In the case where the solutions have integral representations, the Stokes geometries obtained by the Ansatz are consistent with the Stokes phenomen detected by using the method of the steepest descent paths. ・A measure which measures transcendence of a finite subset of a local ring. It is proved that the measure is finite for some family of special functions in the formal power series ring. ・An extension of Gabrielov's theorem to the formal functions is obtained. ・A differential operator acting on the space of pvalent meromorphic functions is introduced and some differential inequalities for the family of pvalent meromorphic functions are obtained by using them. ・For analytic functions on the unit disk that vanish on the origin and the derivative there is identical, distortion inequalities that contains Ruscheweyh derivatives are obtained.

Report
(4results)
Research Output
(26results)