A residue of complex function in three-dimensional vector space

Abstract

In the introduction part of this paper, first of all, the concept of absolute integral sum of complex function is defined, as more general one with respect to the concept of integral as well as of integral sum of "ordinary'' integral calculus. In the main part of the paper, on the basis of redefined concepts of a spatial derivative as well as of a residue of a complex function, the fundamental results of Cauchy's calculus of residues of analytic functions are generalized and the concept of total value of improper integral of complex function is defined. Finally, the results in the form of fundamental lemmas, which are more general with respect to the results of Jordan's fundamental lemmas of Cauchy's calculus of residues of analytic functions, are derived for the general class of scalar complex function whose defining domain is a complex plane.