Value distribution of the Painlevé Transcendents

Abstract

We consider the solutions of the First Painleve Differential Equationω″=z+6w2, commonly known as First Painleve 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−1w(z2), 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 Painleve’s Equations II and IV.