Algebraic Analysis of Exponential Asymptotics

2-paramnetered exponential asymptotic solution to the Dufling equation is constructed by using themultiple-scale analysis.它支持formal solution converges和coincides with theFourier series expansion of the solution written in terms of Jacobi's elliptic function obtained byquadrature. The Stokes geometry of The second and The third Painleve equations are determined. AnAnsatz concerning determination of Stokes gepmetry of third order ordinary differential equationswith 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 the Stokes phenomen detected by the Stokes geometriesusing the method of the steepest descent paths. A measure which measures transcendence of A finitesubset of a local ring. It is proved the measure is finite for some family of special functionsin the formal power series ring. An extension of Gabrielov's theorem to the formal functions isA differential operator acting on the space of p-valent meromorphic functions isintroduced and some differential inequalities for the family of p-valent meromorphic functions areobtained by using them For analytic functions on the unit disk that vanish on the origin and thederivative there is identicaldistortion inequalities that contains Ruscheweyh derivatives are obtained。

・2-paramnetered exponential asymptotic solution to the Dufling equation is constructed by using the multiple-scale 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 p-valent meromorphic functions is introduced and some differential inequalities for the family of p-valent 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.