Frequency of a-points for the fifth and the third Painlevé transcendents in a sector

Shun Shimomura

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

For the fifth Painlevé transcendents in a sector, under the condition that the values taken along some curve tending to infinity are bounded away from 1 and another specified complex number, we present new upper estimates for the number of a-points including poles and for the growth order. As far as we are concerned with the known asymptotic solutions of the fifth Painlevé equation, this condition is easily checked, and our results are applicable to almost all of them. About concrete examples we discuss the frequency of a-points, the equi-distribution property and the growth order. Our method works on the third Painlevé transcendents as well, yielding an analogous result.

Original languageEnglish
Pages (from-to)591-605
Number of pages15
JournalTohoku Mathematical Journal
Volume65
Issue number4
DOIs
Publication statusPublished - 2013

Fingerprint

Sector
Equidistribution
Asymptotics of Solutions
Complex number
Pole
Infinity
Curve
Estimate

Keywords

  • Asymptotic solution
  • Growth order
  • Painlevé equations
  • Value distribution

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Frequency of a-points for the fifth and the third Painlevé transcendents in a sector. / Shimomura, Shun.

In: Tohoku Mathematical Journal, Vol. 65, No. 4, 2013, p. 591-605.

Research output: Contribution to journalArticle

@article{0bf5ffbae78a4e6980e579a4948a8d00,
title = "Frequency of a-points for the fifth and the third Painlev{\'e} transcendents in a sector",
abstract = "For the fifth Painlev{\'e} transcendents in a sector, under the condition that the values taken along some curve tending to infinity are bounded away from 1 and another specified complex number, we present new upper estimates for the number of a-points including poles and for the growth order. As far as we are concerned with the known asymptotic solutions of the fifth Painlev{\'e} equation, this condition is easily checked, and our results are applicable to almost all of them. About concrete examples we discuss the frequency of a-points, the equi-distribution property and the growth order. Our method works on the third Painlev{\'e} transcendents as well, yielding an analogous result.",
keywords = "Asymptotic solution, Growth order, Painlev{\'e} equations, Value distribution",
author = "Shun Shimomura",
year = "2013",
doi = "10.2748/tmj/1386354297",
language = "English",
volume = "65",
pages = "591--605",
journal = "Tohoku Mathematical Journal",
issn = "0040-8735",
publisher = "Tohoku University, Mathematical Institute",
number = "4",

}

TY - JOUR

T1 - Frequency of a-points for the fifth and the third Painlevé transcendents in a sector

AU - Shimomura, Shun

PY - 2013

Y1 - 2013

N2 - For the fifth Painlevé transcendents in a sector, under the condition that the values taken along some curve tending to infinity are bounded away from 1 and another specified complex number, we present new upper estimates for the number of a-points including poles and for the growth order. As far as we are concerned with the known asymptotic solutions of the fifth Painlevé equation, this condition is easily checked, and our results are applicable to almost all of them. About concrete examples we discuss the frequency of a-points, the equi-distribution property and the growth order. Our method works on the third Painlevé transcendents as well, yielding an analogous result.

AB - For the fifth Painlevé transcendents in a sector, under the condition that the values taken along some curve tending to infinity are bounded away from 1 and another specified complex number, we present new upper estimates for the number of a-points including poles and for the growth order. As far as we are concerned with the known asymptotic solutions of the fifth Painlevé equation, this condition is easily checked, and our results are applicable to almost all of them. About concrete examples we discuss the frequency of a-points, the equi-distribution property and the growth order. Our method works on the third Painlevé transcendents as well, yielding an analogous result.

KW - Asymptotic solution

KW - Growth order

KW - Painlevé equations

KW - Value distribution

UR - http://www.scopus.com/inward/record.url?scp=84898859852&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84898859852&partnerID=8YFLogxK

U2 - 10.2748/tmj/1386354297

DO - 10.2748/tmj/1386354297

M3 - Article

AN - SCOPUS:84898859852

VL - 65

SP - 591

EP - 605

JO - Tohoku Mathematical Journal

JF - Tohoku Mathematical Journal

SN - 0040-8735

IS - 4

ER -