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Proceedings of the Edinburgh Mathematical Society (1)

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LOWER ESTIMATES FOR THE GROWTH OF THE FOURTH AND THE SECOND PAINLEVÉ TRANSCENDENTS
Shun Shimomura
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 47 / Issue 1 / February 2004

Published online by Cambridge University Press:

27 May 2004, pp. 231-249
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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