Many-Sheeted Versions of the Plya-Bernstein and Borel Theorems for Entire Functions of Order rho not equal 1 and Their Applications

摘要

The Puiseux series generated by the power function z = w (1/rho), where rho 0,rho not equal 1, is considered. A version of the Plya-Bernstein theorem for an entire function of order rho not equal 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 rho = 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).