TY - JOUR
T1 - Growth results for Painlevé transcendents
AU - Hinkkanen, Aimo
AU - Laine, Ilpo
PY - 2004/11
Y1 - 2004/11
N2 - Painlevé differential equations have been an important topic of research in complex differential equations during the last century, and the last two decades in particular, with many applications not only to pure mathematics but also to physics and engineering. In this paper, we prove that any transcendental solution of the second Painlevé equation w″ = 2w3 + zw + α is of order at least 3/2, and that any transcendental solution of the fourth Painlevé equation 2ww″ = (w′)2 + 3w4 + 8zw3 + 4(z2 - α)w2 + 2β is of order at least 2.
AB - Painlevé differential equations have been an important topic of research in complex differential equations during the last century, and the last two decades in particular, with many applications not only to pure mathematics but also to physics and engineering. In this paper, we prove that any transcendental solution of the second Painlevé equation w″ = 2w3 + zw + α is of order at least 3/2, and that any transcendental solution of the fourth Painlevé equation 2ww″ = (w′)2 + 3w4 + 8zw3 + 4(z2 - α)w2 + 2β is of order at least 2.
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U2 - 10.1017/S0305004104007947
DO - 10.1017/S0305004104007947
M3 - Article
AN - SCOPUS:10044259662
SN - 0305-0041
VL - 137
SP - 645
EP - 655
JO - Mathematical Proceedings of the Cambridge Philosophical Society
JF - Mathematical Proceedings of the Cambridge Philosophical Society
IS - 3
ER -