Value distribution of the fifth Painlevé transcendents in sectorial domains
Every solution of the fifth Painlevé equation is meromorphic on the universal covering of the domain \(\mathbb{C} \setminus \{0\},\) and, in general, it is many-valued around \(x=\infty\) and \(x=0.\) This paper is concerned with a value distribution property for any solution \(y(x)\) in a sector of the form \(| \arg x| <\phi,\) \(r<| x| <R\) as \(R \to\infty\) (respectively, \(r \to 0\)). Let \(N(y, \phi, r, R)\) denote the number of \(1\)-points of \(y(x)\) in this sector. It is shown that \(N(y,\phi, r, R) =O(R^C)\) as \(R\to \infty\) (respectively, \(N(y,\phi, r, R) =O(r^{-C})\) as \(r\to 0\)), where \(C\) is some positive number.
scientific article; zbMATH DE number 5149494