摘要:
推广Riemann P函数的思想(用方程的参数表示方程所定义的函数),引入D函数统一表示正则积分和非正则积分.利用显式解讨论非Fuchs型方程的单值群.得到Floquet解的指标展开系数的显式.根据对应函数法统一研究广义非正则方程的求解问题,包括具有正则和非正则极点,本性奇点,代数,对数和超越奇点以及奇线的方程.利用D函数表示基本解系,从而推广解析理论的研究范围.指出D函数的自守性,并讨论Poincaré猜测的意义.
Abstract:
Extending Riemann's idea of P function (using equation's parameters to represent the function defined by the equation), we introduce correspondence functions D(z) to describe regular and irregular integrals in a unifying way. By explicit solution discuss monodromy group of non-Fuchsian equations. The explicit expressions of exponent and expansion coefficients for Floquet solution are obtained. Method of correspondence functions permits us to obtain systematically the solutions of generalized irregular equations, having regular, irregular poles, essential, algebraic, transcendental, logarithmic singularities as well as singular line. The representation of basic set of solutions by Dσ(z) function makes it possible to enlarge the scope of investigation of analytic theory. The significance of Poincare's conjecture is discussed, as D functions are automorphic.