Supplements To Resurgence Theory
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This chapter is devoted to some general nonsense in resurgence theory which will be useful to study furthermore the first Painlevé equation from the resurgence viewpoint. We define sectorial germs of holomorphic functions (Sect. 7.2) and we introduce the sheaf of microfunctions (Sect. 7.3). This provides an approach to the notion of singularities which is the purpose of Sect. 7.4. We define the formal Laplace transform for microfunctions and for singularities and conversely, the formal Borel transform acting on asymptotic classes (Sect. 7.5). The main properties of the Laplace transform needed in this course are developed to Sect. 7.6. We then introduce some spaces of resurgent functions and define the alien operators (Sect. 7.7 to 7.9).
KeywordsRiemann Surface Holomorphic Function Convolution Product Simple Singularity Constant Sheaf
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