Lower estimates for the growth of the fourth and the second painlevé transcendents

Shun Shimomura

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Let w(z) be an arbitrary transcendental solution of the fourth (respectively, second) Painlevé equation. Concerning the frequency of poles in |z| ≤ r, it is shown that n(r,w) ≫ r2 (respectively, n(r,w) ≫ r3/2), from which the growth estimate T(r,w) ≫ r 2 (respectively, T(r,w) ≫ r3/2) immediately follows.

Original languageEnglish
Pages (from-to)231-249
Number of pages19
JournalProceedings of the Edinburgh Mathematical Society
Issue number1
Publication statusPublished - 2004 Feb 1



  • Characteristic function
  • Elliptic functions
  • Growth order
  • Painlevé transcendents

ASJC Scopus subject areas

  • Mathematics(all)

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