### Abstract

For the first Painlevé equation, it is proved that every meromorphic solution satisfies T(r, w) = O(r^{5/2}). In showing this estimate, we employ two types of auxiliary function, one of which is crucial in the proof of the Painlevé property. Our method is also applicable to the second (resp. the fourth) Painlevé transcendents, and we obtain T(r, w) = O(r^{3}) (resp. O(r^{4})).

Original language | English |
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Pages (from-to) | 259-269 |

Number of pages | 11 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 134 |

Issue number | 2 |

Publication status | Published - 2003 Mar |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Proceedings of the Cambridge Philosophical Society*,

*134*(2), 259-269.

**Growth of the first, the second and the fourth Painlevé transcendents.** / Shimomura, Shun.

Research output: Contribution to journal › Article

*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 134, no. 2, pp. 259-269.

}

TY - JOUR

T1 - Growth of the first, the second and the fourth Painlevé transcendents

AU - Shimomura, Shun

PY - 2003/3

Y1 - 2003/3

N2 - For the first Painlevé equation, it is proved that every meromorphic solution satisfies T(r, w) = O(r5/2). In showing this estimate, we employ two types of auxiliary function, one of which is crucial in the proof of the Painlevé property. Our method is also applicable to the second (resp. the fourth) Painlevé transcendents, and we obtain T(r, w) = O(r3) (resp. O(r4)).

AB - For the first Painlevé equation, it is proved that every meromorphic solution satisfies T(r, w) = O(r5/2). In showing this estimate, we employ two types of auxiliary function, one of which is crucial in the proof of the Painlevé property. Our method is also applicable to the second (resp. the fourth) Painlevé transcendents, and we obtain T(r, w) = O(r3) (resp. O(r4)).

UR - http://www.scopus.com/inward/record.url?scp=0037365285&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037365285&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0037365285

VL - 134

SP - 259

EP - 269

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 2

ER -