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The Stokes phenomenon in exact asymptotics. (English) Zbl 0952.34074

The paper begins with a new and elementary constructive proof for the multisummability properties of formal solutions to linear ordinary differential equations at irregular singular points. This then serves as a model to illustrate the geometric approach to multisummation. Several basic properties of multisums and the associated sheaves are derived. The paper then moves on to study the Cauchy-Heine transforms in relation to multisummation and the Stokes phenomenon. The authors then show how to construct multisums with a prescribed Stokes phenomenon using the Malgrange-Sibuya isomorphism. Starting with the Stokes automorphisms the authors introduce the alien derivations of J. Ecalle and derive the Ecalle’s bridge equation for the general integral linear ordinary differential equations. All the main ideas are clearly illustrated with some very lucid examples.

MSC:

34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain
30D10 Representations of entire functions of one complex variable by series and integrals
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