Series expansions of painlevé transcendents near the point at infinity

Shun Shimomura

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


For the Painlevé equations (I) through (V) near the point at infinity we present several families of two-parameter solutions. Our solutions are expressed by asymptotic series with coefficients polynomial in exponential terms, and also by convergent power series in exponential terms with coefficients expanded into asymptotic series. Both expressions are valid without a restriction on integration constants. We propose a direct method to derive asymptotic solutions, which is also applicable to more general nonlinear equations. As applications of our results, for general solutions of the third and the fifth Painlevé equations, we give estimates for the number of a-points including poles in given sectors.

Original languageEnglish
Pages (from-to)277-319
Number of pages43
JournalFunkcialaj Ekvacioj
Issue number2
Publication statusPublished - 2015 Aug 13


  • Asymptotic solutions
  • Hamiltonian system
  • Painlevé equations

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology


Dive into the research topics of 'Series expansions of painlevé transcendents near the point at infinity'. Together they form a unique fingerprint.

Cite this