### 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) ≫ r^{2} (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 |
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Pages (from-to) | 231-249 |

Number of pages | 19 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 47 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 Feb |

### Fingerprint

### Keywords

- Characteristic function
- Elliptic functions
- Growth order
- Painlevé transcendents

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Lower estimates for the growth of the fourth and the second painlevé transcendents.** / Shimomura, Shun.

Research output: Contribution to journal › Article

*Proceedings of the Edinburgh Mathematical Society*, vol. 47, no. 1, pp. 231-249. https://doi.org/10.1017/S0013091503000440

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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 -