Martin Berz
berz@msu.edu
Department of Mathematics and
Department of Physics and Astronomy
Michigan State University
East Lansing, MI 48824, USA
We develop the basic elements of a Cauchy theory on the complex Levi-Civita field, which constitutes the smallest algebraically closed non-Archimedean extension of the complex numbers. We introduce a concept of analyticity based on differentiation. We show that analytic functions can be integrated over suitable piecewise smooths paths in the sense of integrals developed in an accompanying paper. It is then shown that the resulting path integrals allow the formulation of a workable Cauchy theory in a rather similar way as in the conventional case. In particular, we obtain a Cauchy theorem, the Cauchy formulas for the function and its derivatives, as well as analogues of several other of the conventional theorems.