非Fuchs型方程的新理论——树级数解的表现定理(Ⅰ)

在微分方程的解析理论中非Fuchs型方程的严格显式解至今并未求得(Poincaré问题),本文提出的新理论首次给出非正则积分的一般求法和显式的精确解. 本法与经典理论的根本不同在于摈弃形式解的假定,从方程本身建立对应关系,应用留数定理自动给出非正则积分的解析结构.它由无收缩部和全、半收缩部组成.前者是通常的递推级数,后者则表为树级数.树级数是类新颖的解析函数,通常的递推级数只是它的特例而已. 本文的目的是建立非正则积分的一般理论,为此需要阐明Poincaré问题(1880T.I.P.333)的实质1]:无法求出非正则积分的显式.根据以下证明的表现定理, 非正则积分是类新颖的解析函数,其中系数D_nk是方程参数的常项树级数.

New Theory for Equations of Non-Fuchsian Type Representation Theorem of Tree Series Solution(I)

In the analytic theory of differential equations the exact explicit analytic solution has not been obtained for equations of the non-Fuchsian type(Poincare's problem). The new theory proposed in this paper for the first time affords a general method of finding exact analytic expres-sion for irregular integrals.By discarding the assumption of formal solution of classical theory,our method consists in deriving a cor-respondence relation from the equation itself and providing the analytic structure of irregular integrals naturally by the residue theorem. Irregular integrals are made up of three parts: noncontracted part,represented by ordinary recursion series,all-and semi-contracted part by the so-called tree series. Tree series solutions belong to analytic function of the new kind with recursion series as the special case only.