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LOWER ESTIMATES FOR THE GROWTH OF THE FOURTH AND THE SECOND PAINLEVÉ TRANSCENDENTS

Published online by Cambridge University Press:  27 May 2004

Shun Shimomura
Shun Shimomura
Affiliation:
Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan (shimomur@math.keio.ac.jp)
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Abstract

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Let $w(z)$ be an arbitrary transcendental solution of the fourth (respectively, second) Painlevé equation. Concerning the frequency of poles in $|z|\le r$, it is shown that $n(r,w)\gg r^2$ (respectively, $n(r,w)\gg r^{3/2}$), from which the growth estimate $T(r,w)\gg r^2$ (respectively, $T(r,w)\gg r^{3/2}$) immediately follows.

AMS 2000 Mathematics subject classification: Primary 34M55; 34M10

Keywords

Painlevé transcendentsgrowth ordercharacteristic functionelliptic functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 1 , February 2004 , pp. 231 - 249
Copyright
Copyright © Edinburgh Mathematical Society 2004