Real Version of Calculus of Complex Variable (II): Cauchy's Point of View

TL;DR

This paper presents a real-variable approach to complex calculus using harmonic differential forms in the Kähler algebra, paralleling classical Cauchy theory without new differentiation concepts.

Contribution

It introduces a simplified real-variable framework for complex calculus based on harmonic differential forms satisfying Cauchy-Riemann relations.

Findings

Power series are established in the real-variable setting.

Residue theorem is derived within this framework.

Approach simplifies complex calculus concepts.

Abstract

As was the case in a previous paper, the differential form x+ydxdy plays the role that the variable z plays in the standard calculus of complex variable. The role of holomorphic functions will now be played by strict harmonic differential forms in the Kaehler algebra of the real plane. These differential forms satisfy the Cauchy-Riemann relations. No new concept of differentiation is needed, and yet this approach parallels standard Cauchy theory, but more simply. The power series and theorem of residues belong here at the end, unlike in the previous paper.