overconvergent

Overconvergent is a term used in mathematics, particularly in p-adic analysis and analytic number theory, to describe certain types of power series. A power series is considered overconvergent if it converges not only on a disk but also in a larger region or with a faster rate of convergence than what is typically implied by its formal definition. This concept is crucial for extending the analytic properties of functions defined by power series beyond their initial radius of convergence.

The notion of overconvergence often arises when dealing with analytic functions in the context of p-adic numbers. In this setting, power series can exhibit unusual convergence behavior compared to their complex analytic counterparts. Overconvergent series are those that can be extended to a larger domain than their formal radius of convergence would suggest, or whose coefficients decay sufficiently rapidly to ensure convergence in a broader sense. This allows mathematicians to study the behavior of functions in a more robust manner, enabling deeper insights into their structure and properties. Overconvergence is a key tool in the study of p-adic analytic spaces and in constructing certain categories of functions in p-adic geometry.