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Frequency of \(a\)-points for the fifth and the third Painlevé transcendents in a sector. (English) Zbl 1305.34152

In this paper value distribution properties for solutions of the fifth and the third Painlevé equations in a sector are studied. Supposing that the values taken along some curve tending to infinity are bounded away from 1 and another specified complex number, the author obtains new good upper estimates, which are expected to be the best possible for general solutions, for the number of \(a\)-points (i.e., points where a solution in question takes the value \(a\); \(a\) is allowed to be infinity, that is, the number of poles are also included) and for the growth order in a sector containing the curve. The main results are applicable to almost all known asymptotic solutions of the fifth Painlevé equation: Using the main results, the author discusses also the frequency of \(a\)-points, the equi-distribution property, and the growth order for several concrete asymptotic solutions as well.

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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References:

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