Resurgent function

Summary

Resurgent functions are a class of analytic functions introduced by Jean Écalle to study divergent series. A resurgent series is a formal series whose Borel transform converges near $0$. This Borel transform must also continue analytically along every path that avoids a discrete set of singular points. Many divergent series of analysis are resurgent, such as Stirling's series and the formal solutions of many analytic differential equations. The singularities in the Borel plane link the divergent series to exponentially small terms. This is the base of Borel summation beyond its simplest cases.