A Step Beyond Borel-Laplace Summability

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


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.


Short Path Riemann Surface Holomorphic Function Connected Domain Homotopy Class 
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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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