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)

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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.",

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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.

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KW - Elliptic functions

KW - Growth order

KW - Painlevé transcendents

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