Abstract
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.