Real Version of Calculus of Complex Variable (I): Weierstrass Point of   View
  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. 
Jose G. Vargas
math.GM