Real Version of Calculus of Complex Variable (I): Weierstrass Point of View

TL;DR

This paper develops a real-plane version of complex variable calculus using Kähler algebra, simplifying key theorems and formulas without relying on complex differentiation, and offers a new perspective aligned with Weierstrass's approach.

Contribution

It introduces a real algebraic framework for complex analysis concepts, replacing traditional complex differentiation with Clifford algebra methods and algebraic integration.

Findings

Derivation of Cauchy's integral formulas in real algebraic form

Establishment of a real-plane residue theorem for multiply connected regions

Simplification of complex variable theory for physicists

Abstract

A very small amount of K\"ahler algebra (i.e. Clifford algebra of differential forms) in the real plane makes x + ydxdy emerge as a factor between the differentials of the Cartesian and polar coordinates, largely replacing the concept of complex variable. The integration on closed curves of closed 1-forms on multiply connected regions takes us directly to a real plane version of the theorem of residues. One need not resort to anything like differentiation and integration with respect to x + ydxdy. It is a matter of algebra and integration of periodic functions. We then derive Cauchy's integral formulas, including the ones for the derivatives. Additional complex variable theory of general interest for phyicists is then trivial. The approach is consistent with the Weierstrass point of view: power series expansions, even if explicit expressions are not needed. By design, this approach cannot replace integrations that yield complex results. These can be obtained with an approach based on the Cauchy point of view, where the Cauchy-Riemann conditions come first and the theorem of residues comes last.