TY - JOUR
T1 - Lower estimates for the growth of the fourth and the second painlevé transcendents
AU - Shimomura, Shun
N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.
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
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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 -