The indicator diagram theory for maximal type entire functions and the characterization of analytic continuability

Abstract

We extend Pólya's indicator diagram theory to encompass entire functions of order at most 1, allowing functions of maximal type. To do so, we introduce an extension of the complex plane in which indicator diagrams may be unbounded or even purely infinite. In this framework, we establish the Laplace and inverse Laplace transforms, and prove an analog of Pólya's theorem. As an application, we extend a characterization of the analytic continuability of power series. The resulting theory is illustrated by a range of classical examples of special functions.