Value distribution of the Painlevé transcendents
type: Item
instance of: scholarly article
title: Value distribution of the Painlevé transcendents
zbMATH Open document ID: 1016.34091
DOI: 10.1007/BF02785417
author: Norbert Steinmetz
published in: Israel Journal of Mathematics
publication date: 2002-10-23T00:00:00Z
review text: The order of growth of the first Painlevé transcendents is studied. For a regular point \(z_0\neq 0\) of a fixed solution \(w(z)\) to the first Painlevé differential equation \(w''= z+6w^2\), the author defines the main working tool, the so-called scaling unit \[ r(z_0)= \min\{|w(z_0)|^{-1/2}, |w'(z_0)|^{-1/3}, |z_0|^{-1/4}\}. \] With help of this notion, some form of the theorem on analytic dependence on parameters and initial values is proved. Using these results, the author obtains the following asymptotic relations for the counting function \(N(r,w)= O(r^{5/2})\), for the Nevanlinna characteristic function \(T(r,w)= O(r^{5/2})\) and the proximity function \(m(r,w)= O(\log r)\), where \(w(z)\) is any first Painlevé transcendent. These relations yield, that the first Painlevé transcendent has the order of growth \[ \lambda(\omega)= \limsup_{r\to\infty} {\log T(r,w)\over\log r}={5\over 2}. \]
reviewed by: A. Klíč
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 34M10
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 1819979
zbMATH Keywords: Painlevé transcendents
zbMATH Keywords: order of growth
zbMATH Keywords: spherical derivative
zbMATH Keywords: Nevanlinna characteristic function
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02785417
OpenAlex ID: W2137950485
description: scientific article; zbMATH DE number 1819979