Bemerkungen über elementare Funktionen in nichtarchimedischen Banachalgebren

This paper deals with the behaviour of power series with coefficients in a non-archimedean valued field Ω for arguments in a non-archimedean Banachalgebra E/Ω. Beyond a trivial augmentation, the domain of convergence is extended in relation to Ω by certain elements b ∈ E for which \(\mathop {\lim }\limits_{n \to \infty } \left( {\frac{{||b^n ||}}{{||b||^n }}} \right) = 0\) . These elements are called pseudo-nilpotent and characterized by \(\mathop {\lim }\limits_{n \to \infty } (||b^n ||^{\frac{1}{n}} )< ||b||\) . The examples of exponential functions, logarithms and powers show the changes in relation to the methods and results in Ω caused by the extension of the domain of convergence and the absence of norm multiplicativity and invertibility in E. Finally the algebraic-topological structure of a part of the domain of convergence of the exponential function is presented.