Value distribution of the third Painlevé transcendents in sectorial domains

Abstract

This article concerns the value distribution of the third Painlevé transcendents in sectorial domains around fixed singular points. We show that the cardinality of the zeros of a third Painlevé transcendent in a sector has an asymptotic growth of finite order, thereby giving an improvement of the known estimation.

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…”