Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 17-49

A UNIFIED APPROACH TO THE PAINLEVÉ TRANSCENDENTS

Norbert Steinmetz

Technische Universität Dortmund, Fakultät für Mathematik
D-44221 Dortmund, Germany; norbert.steinmetz 'at' mathematik.uni-dortmund.de

Abstract. We utilise a recent approach via the so-called re-scaling method to derive a unified and comprehensive theory of the solutions to Painlevé's differential equations (I), (II) and (IV), with emphasis on the most elaborate equation (IV).

2010 Mathematics Subject Classification: Primary 34M05, 30D35, 34M55, 30D45.

Key words: Stokes sector, Stokes ray, asymptotic expansion, re-scaling, pole-free sector, Airy equation, Weber–Hermite equation, sub-normal solution.

Reference to this article: N. Steinmetz: A unified approach to the Painlevé transcendents. Ann. Acad. Sci. Fenn. Math. 42 (2017), 17-49.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4203