The publication combines the beneficial properties of a graduate-level textbook with these of a examine monograph and survey of the new effects on research and geometry of differential equations within the actual and intricate area. As a graduate textbook, it contains self-contained, occasionally significantly simplified demonstrations of numerous basic effects, which formerly seemed simply in magazine courses (desingularization of planar analytic vector fields, lifestyles of analytic separatrices, confident and unfavorable effects at the Riemann-Hilbert challenge, Ecalle-Voronin and Martinet-Ramis moduli, answer of the Poincare challenge at the measure of an algebraic separatrix, etc.). As a examine monograph, it explores in a scientific means the algebraic decidability of neighborhood category difficulties, stress of holomorphic foliations, and so on. every one part ends with a set of difficulties, partially meant to assist the reader to realize knowing and event with the fabric, in part drafting demonstrations of the more moderen effects surveyed within the textual content. The exposition of the e-book is usually geometric, although the algebraic facet of the structures can be prominently featured. On numerous events the reader is brought to adjoining components, comparable to intersection thought for divisors at the projective aircraft or geometric thought of holomorphic vector bundles with meromorphic connections. The publication presents the reader with the primary instruments of the trendy thought of analytic differential equations and intends to function a regular resource for references during this region.

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