  The present article is focused on the study of a special class of systems of
nonlinear transcendental equations for which classical algebraic and symbolic
methods are inapplicable. For the purpose of the study of such systems, we
develop a method for computing residue integrals with integration over certain
cycles. We describe conditions under which the mentioned residue integrals
coincide with power sums of the inverses to the roots of a system of equations
(i.e., multidimensional Waring's formulas). As an application of the suggested
method, we consider a problem of finding sums of multi-variable number series.
