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Value distribution of the Painlevé Transcendents

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Abstract

We consider the solutions of the First Painlevé Differential Equationω″=z+6w 2, commonly known as First Painlevé Transcendents. Our main results are the sharp order estimate λ(w)≤5/2, actually an equality, and sharp estimates for the spherical derivatives ofw andf(z)=z −1 w(z 2), respectively:w#(z)=O(|z|3/4) andf#(z)=O(|z|3/2). We also determine in some detail the local asymptotic distribution of poles, zeros anda-points. The methods also apply to Painlevé’s Equations II and IV.

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  1. Fachbereich Mathematik, Universität Dortmund, D-44221, Dortmund, Germany

    Norbert Steinmetz

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  1. Norbert Steinmetz
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Steinmetz, N. Value distribution of the Painlevé Transcendents. Isr. J. Math. 128, 29–52 (2002). https://doi.org/10.1007/BF02785417

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  • DOI: https://doi.org/10.1007/BF02785417

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