Poincaré非正则积分问题的新进展
New Development in Poincaré’s Problem of Irregular Integrals
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摘要: 关于非Fuchs型方程,Poincaré曾经作出过重要的论断:没有方法可以求出非正则积分的显示表述.为了阐明这一论断的实质,我们证明对应定理:非正则积分是类具有树结构的新型解析函数,其中一部份解是通常的递推级数,而另一部则是不遵循递推关系的“树级数”.与经典理论(Hill-Poincaré-von Koch)计算无穷行列式的数值解不同,本法自然地给出严格解析解的显示表式.本法可以建立统一的解析理论以讨论一般变系数方程,包括有奇线在内的多种奇点.由于树级数具有自守性.我们讨论Poincaré猜测的意义.Abstract: In connection with non-Fuchsian equations Poincaré has made an important conclusion; It is impossible to obtain explicit expressions of irregular integrals. To elucidate the essence of Poincaré's problem, we establish correspondence theorem, Irregular integrals are analytic functions of new kind, possessing tree structure, part of which can be represented by conventional recursive series, while its remaining part is expressed by the so-called tree series, not subjecting to any recursive relation at all. In contrast to the numerical solution calculated by infinite determinant of classical theory (Hill-Poincaré-von Koch),our method yields naturally exact analytic solution in explicit form, The method proposed map be used to;construct a unifying theory for general equations with variable coefficients, having varioas kinds of singularities as singular lines. The significance of Poincaré conjecture is discussed. The tree series obtained belong to higher automorphic functions.
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