Abstract:

In this article we derive all salient properties of analytic functions, including the analytic version of the inverse function theorem, using only the most elementary convergence properties of series. Not even the notion of differentiability is required to do so. Instead, analytical arguments are replaced by combinatorial arguments exhibiting properties of formal power series. Along the way, we show how formal power series can be used to solve combinatorial problems and also derive some results in calculus with a minimum of analytical machinery.

Keywords: formal power series; analytic functions; analytic inverse function theorem; combinatorial and analytical aspects

Document Type: Research article

DOI: 10.1080/00207390701784577

Affiliations: 1: Arbeitsgruppe Mathematik, Fachhochschule Wiesbaden, Wiesbaden, Germany