This monograph should be of interest to a broad spectrum of readers: specialists in discrete and continuous mathematics, physicists, engineers, and others interested in computing sums and applying complex analysis in discrete mathematics. It contains investigations on the problem of finding integral representations for and computing finite and infinite sums (generating functions); these arise in practice in combinatorial analysis, the theory of algorithms and programming on a computer, probability theory, group theory, and function theory, as well as in physics and other areas of knowledge. A general approach is presented for computing sums and other expressions in closed form by reducing them to one-dimensional and multiple integrals, most often to contour integrals.
1. The method of integral representation of combinatorial sums; 2. Integral representation and computation of one-dimensional combinatorial sums; 3. Inversion and classification of linear relations in combinatorial analysis; 4. Combinatorial interpretation, integral representation, and estimation of certain sums in combinatorial analysis; 5. Integral representation and computation of multi-dimensional sums; 6. Applications; 7. Open problems.