Value distribution of the fifth Painlevé transcendents in sectorial domains

Abstract

This article is concerned with a value distribution of the fifth Painlevé transcendents in sectorial domains around a fixed singular point. We show that the cardinality of the 1-points of a fifth Painlevé transcendent in a sector has an asymptotic growth of finite order, thereby giving an improvement of the known estimates.

Equi-Distribution of Values for the Third and the Fifth Painlevé Transcendents

“…We remark that the constant Λ in (5.3) can be chosen so that it is independent of α, β, γ, δ, because (5.3) is considered in the sector ( [8], [9]). Since (γ, δ) = (0, 0), using Remark 2.3, Lemma 4.5 and (5.2), we have…”

Asymptotic behaviour of the fifth Painlevé transcendents in the space of initial values

“…However, the literature on the asymptotic behaviours of the fifth Painlevé transcendent concentrates on behaviours on the real line, often focusing on special behaviours or solutions, while we consider all solution behaviours for . For other mathematical results related to (see , while for applications in physics see , and references therein).…”

“…The asymptotic analysis of the fifth Painlevé transcendent has been studied by many authors, including [2-4, 6, 29, 30, 34-36, 42, 47], and [24]. However, the literature on the asymptotic behaviours of the fifth Painlevé transcendent concentrates on behaviours on the real line, often focusing on special behaviours or solutions, while we consider all solution behaviours for x ∈ C. For other mathematical results related to P V (see [5,7,16,17,28,31,43,46], while for applications in physics see [11,25,44], and references therein).…”

“…The asymptotic analysis of the fifth Painlevé transcendent has been studied by many authors, including Andreev and Kitaev (1997a, 1997b, 2000, Zeng and Zhao (2016), Bryuno and Parus and Qin and Shang (2006), McLeod (1999a, 1999b), McCoy and Tang (1986a, 1986b, 1986c, and Jimbo (1982). However, the literature on the asymptotic behaviours of the fifth Painlevé transcendent concentrates on behaviours on the real line, often focusing on special behaviours or solutions, while we consider all solution behaviours for x ∈ C. For other mathematical results related to P V , see (Boelen et al, 2013;Shimomura, 2011;Kaneko and Ohyama, 2007;Sasaki, 2007;Clarkson, 2005;Lu and Shao, 2004;Gordoa et al, 2001a;2001b), while for applications in physics see (Jimbo et al, 1980;Dyson, 1995;Schief, 1994), and references therein.…”