Supplements To Resurgence Theory

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 pro...

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Published inDivergent Series, Summability and Resurgence III Vol. 2155; pp. 147 - 207
Main Author Delabaere, Eric
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 01.01.2016
Springer International Publishing
SeriesLecture Notes in Mathematics
Subjects
Calculus & mathematical analysis
Complex analysis
Constant Sheaf
Convolution Product
Differential equations
Holomorphic Function
Riemann Surface
Simple Singularity
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Summary: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).
ISBN:9783319289991
3319289993
ISSN:0075-8434
1617-9692
DOI:10.1007/978-3-319-29000-3_7