Divergent series and differential equations
Divergent series and differential equations
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发散级数和微分方程
DOI：
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发表时间：
2014
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通讯作者：
M. Loday
中科院分区：
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作者：
M. Loday
我们发展了已知的各种方法来研究一类包含复域中常微分方程的所有发散级数解的级数的可和性。我们首先研究了散度仅依赖于一个参数（水平k或临界时间）的情况，称为k-可和性。我们研究了当散度依赖于称为多重可和性的几个级别时的推广情况。证明了这些定义的相干性及其等价性，并给出了一些应用。我们还提供了关于Gevrey渐近性的必要基础知识，并概述了轴理论、上同调和线性常微分方程。研究了各种各样的例子，包括复平面上的微分同构的切同胚芽的例子。
We develop the various known approaches to the summability of a class of series that contains all divergent series solutions of ordinary differential equations in the complex field. We first study the case when the divergence depends only on one parameter (the level k or critical time) called k-summability. We study then generalizations to the case when the divergence depends on several levels called multi-summability. We prove the coherence of the definitions and their equivalences and we provide some applications. We also provide the necessary basics on Gevrey asymptotics and a survey of sheaf theory, cohomology and linear ordinary differential equations. Various examples are worked on, including the example of tangent-to-identity germs of diffeomorphisms in the complex plane.