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In the introductory part of this paper, a notion of absolute integral sums of a complex function, which is more general with respect to that of an integral and integral sums of ordinary integral calculus, is defined. Throughout the main part of the paper, an attempt has been made to generalize, on the basis of redefining the notion of a complex function residue, some of the fundamental results of Cauchy's calculus of residues of analytic functions. The foundation stone of the whole theory is the total value of an improper integral of complex functions

On a residue of complex functions in the three-dimensional Euclidean complex vector space