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A Step Beyond Borel-Laplace Summability

  • Eric Delabaere
Chapter
  • 803 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 2155)

Abstract

We previously showed that the minor \(\widehat{W}\) of the unique formal series solution \(\widetilde{W}\) of the prepared ODE associated with the first Painlevé equation, defines a function holomorphic on a cut plane. We further analyze the analytic properties of \(\widehat{W}\). We show in Sect. 4.5 how \(\widehat{W}\) can be analytically continued to a domain of a Riemann surface, defined in Sect. 4.2, and we draw some consequences. This question is related to the problem of mastering the analytic continuations of convolution products and, as a byproduct, of getting qualitative estimates on any compact set. This is what we will partly do in Sect. 4.3 and Sect. 4.4, using only elementary geometrical arguments. We end with some supplements in Sect. 4.6.

Keywords

Short Path Riemann Surface Holomorphic Function Connected Domain Homotopy Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Eric Delabaere
    • 1
  1. 1.Département de MathématiquesUniversité d’AngersAngersFrance

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