Lower estimates for the growth of the fourth and the second painlevé transcendents

Shun Shimomura

doi:10.1017/S0013091503000440

Lower estimates for the growth of the fourth and the second painlevé transcendents

Shun Shimomura

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

Let w(z) be an arbitrary transcendental solution of the fourth (respectively, second) Painlevé equation. Concerning the frequency of poles in |z| ≤ r, it is shown that n(r,w) ≫ r² (respectively, n(r,w) ≫ r^3/2), from which the growth estimate T(r,w) ≫ r ² (respectively, T(r,w) ≫ r^3/2) immediately follows.

Original language	English
Pages (from-to)	231-249
Number of pages	19
Journal	Proceedings of the Edinburgh Mathematical Society
Volume	47
Issue number	1
DOIs	https://doi.org/10.1017/S0013091503000440
Publication status	Published - 2004 Feb

Keywords

Characteristic function
Elliptic functions
Growth order
Painlevé transcendents

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.1017/S0013091503000440

Fingerprint Dive into the research topics of 'Lower estimates for the growth of the fourth and the second painlevé transcendents'. Together they form a unique fingerprint.

Cite this

@article{2ed0d42e81c14b208731b11a5c10555e,

title = "Lower estimates for the growth of the fourth and the second painlev{\'e} transcendents",

abstract = "Let w(z) be an arbitrary transcendental solution of the fourth (respectively, second) Painlev{\'e} equation. Concerning the frequency of poles in |z| ≤ r, it is shown that n(r,w) ≫ r2 (respectively, n(r,w) ≫ r3/2), from which the growth estimate T(r,w) ≫ r 2 (respectively, T(r,w) ≫ r3/2) immediately follows.",

keywords = "Characteristic function, Elliptic functions, Growth order, Painlev{\'e} transcendents",

author = "Shun Shimomura",

year = "2004",

month = feb,

doi = "10.1017/S0013091503000440",

language = "English",

volume = "47",

pages = "231--249",

journal = "Proceedings of the Edinburgh Mathematical Society",

issn = "0013-0915",

publisher = "Cambridge University Press",

number = "1",

}

TY - JOUR

T1 - Lower estimates for the growth of the fourth and the second painlevé transcendents

AU - Shimomura, Shun

PY - 2004/2

Y1 - 2004/2

N2 - Let w(z) be an arbitrary transcendental solution of the fourth (respectively, second) Painlevé equation. Concerning the frequency of poles in |z| ≤ r, it is shown that n(r,w) ≫ r2 (respectively, n(r,w) ≫ r3/2), from which the growth estimate T(r,w) ≫ r 2 (respectively, T(r,w) ≫ r3/2) immediately follows.

AB - Let w(z) be an arbitrary transcendental solution of the fourth (respectively, second) Painlevé equation. Concerning the frequency of poles in |z| ≤ r, it is shown that n(r,w) ≫ r2 (respectively, n(r,w) ≫ r3/2), from which the growth estimate T(r,w) ≫ r 2 (respectively, T(r,w) ≫ r3/2) immediately follows.

KW - Characteristic function

KW - Elliptic functions

KW - Growth order

KW - Painlevé transcendents

UR - http://www.scopus.com/inward/record.url?scp=2442609616&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2442609616&partnerID=8YFLogxK

U2 - 10.1017/S0013091503000440

DO - 10.1017/S0013091503000440

M3 - Article

AN - SCOPUS:2442609616

VL - 47

SP - 231

EP - 249

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 1

ER -

Lower estimates for the growth of the fourth and the second painlevé transcendents

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Cite this