On an alternative view to complex calculus

Abstract

In most (if not all) textbooks on complex calculus, the differentiation and integration of complex functions are presented by using the algebraic form of complex variables because the respective formulae in terms of the polar form are inappropriate. In this paper, we demonstrate that by transferring the field structure of the system of complex numbers to the Riemann surface of complex logarithm and changing the sense of derivative and integral, complex calculus can be delivered in terms of the polar form of complex variables identically to the presentation in terms of algebraic form.