Riemann surface of complex logarithm and multiplicative calculus

TL;DR

This paper proposes an elementary approach to complex analysis using multiplicative calculus and Riemann surfaces, simplifying concepts like the logarithm and residues for educational purposes.

Contribution

It introduces a reoriented elementary complex analysis framework that simplifies key concepts by employing multiplicative calculus and Riemann surfaces, avoiding advanced mathematical tools.

Findings

Single-valued logarithmic and power functions achieved

Cauchy integral formula integrated into fundamental theorem of calculus

Residues and singularities are effectively removed

Abstract

Could elementary complex analysis, which covers the topics such as algebra of complex numbers, elementary complex functions, complex differentiation and integration, series expansions of complex functions, residues and singularities, and introduction to conformal mappings, be made more elementary? In this paper we demonstrate that a little reorientation of existing elementary complex analysis brings a lot of benefits, including operating with single-valued logarithmic and power functions, making the Cauchy integral formula as a part of fundamental theorem of calculus, removal of residues and singularities, etc. Implicitly, this reorientation consists of resolving the multivalued nature of complex logarithm by considering its Riemann surface. But instead of the advanced mathematical concepts such as manifolds, differential forms, integration on manifolds, etc, which are necessary for introducing complex analysis in Riemann surfaces, we use rather elementary methods of multiplicative calculus. We think that such a reoriented elementary complex analysis could be especially successful as a first course in complex analysis for the students of engineering, physics, even applied mathematics programs who indeed do not see a second and more advanced course in complex analysis. It would be beneficial for the students of pure mathematics programs as well because it is more appropriate introduction to complex analysis on Riemann surfaces rather than the existing one.