P-adic Line Integrals and Cauchy's Theorems

TL;DR

This paper develops a p-adic analog of complex line integrals, leading to new versions of classical theorems like Cauchy's, and explores their applications to rational, Krasner analytic, and special functions, including L-functions.

Contribution

It introduces a novel p-adic line integral framework that extends classical complex analysis theorems to the p-adic setting, with computational applications.

Findings

Established p-adic versions of Residue and Cauchy-Goursat theorems.

Computed integrals for various classes of functions, including rational and Krasner analytic functions.

Produced values of Kubota-Leopoldt L-functions at integers.

Abstract

Working in the p-adic analog of the complex numbers, we'll define a line integral on a small arc of a circle. This allows new versions of the Residue Theorem, the Cauchy-Goursat Theorem on discs with and without holes, Cauchy's Integral Formula and the Z-P Theorem. In contrast to results in complex analysis, these integrals allow the points on a boundary circle, the bulk of a p-adic disc, to be treated the same as points interior to the boundary circle. The theory of the integral is developed, especially for functions holomorphic on an open disc, and integrals will be calculated for rational functions, Krasner analytic functions and some well-known functions that are not Krasner analytic. Some computations will produce values of Kubota-Leopoldt L-functions at ordinary integers.