Value distribution of the Painlevé Transcendents

Book Review: Painlevé differential equations in the complex plane

“…Only recently estimates of the orders of growth and also the asymptotic distribution of poles, already stated by Boutroux, were confirmed (see [Sh] and [St2]). …”

“…It reports on recent results achieved by the third author [Sh] and the reviewer [St2], independently.…”

Meromorphic solutions of P_4,34 and their value distribution

“…Their solutions are meromorphic functions in the sense that every local solution has a continuation to a function meromorphic in C. For recent proofs see Hinkkanen and Laine in [10] for P 2 or Steinmetz in [29] for both equations. The solutions are also known to be of finite order [26,27,30]. The deficiencies and ramification indices of solutions have been estimated both in case of P 2 and P 4 .…”

Equi-Distribution of Values for the Third and the Fifth Painlevé Transcendents

“…( [7], [14], [15], [19]), which together with the well-known Clunie lemma implies m(r, a, w I ) = O(log r) for every a ∈ C ∪ {∞}. Observing that log N (r, a, w I ) − log T (r, w I ) = log 1 − m(r, a, w…”

“…+ (w − 1) 2 αw + β w + γe z w + δe 2z w(w + 1) w − 1 , whose solutions are meromorphic in C. For solutions of (I), (II) and (IV), equi-distribution properties of values immediately follow from the finiteness of their growth order ( [1], [13], [14], [19]). Each solution w I (z) of (I) is transcendental, and satisfies ̺(w I ) = 5/2 …”