Many-Sheeted Versions of the Pólya–Bernstein and Borel Theorems for Entire Functions of Order ρ ≠ 1 and Their Applications

Abstract

The Puiseux series generated by the power function z = w^1/ρ, where ρ > 0,ρ ≠ 1, is considered. A version of the Pólya–Bernstein theorem for an entire function of order ρ ≠ 1 and normal type is proposed and applied to describe the domain of analytic continuation of this series. The domain of summability of a “regular” Puiseux series is found (this is a many-sheeted “Borel polygon”); in the case ρ = 1, the “one-sheeted” result of Borel is substantially extended. These results make it possible to describe domains of analytic continuation of the Puiseux expansions of popular many-sheeted functions (such as inverses of rational functions).