THE MATHEMATICS OF PRINCIPAL VALUE INTEGRALS AND APPLICATIONS TO NUCLEAR-PHYSICS, TRANSPORT-THEORY, AND CONDENSED MATTER PHYSICS

A review of developments in the mathematics and methods for principal value (PV) integrals is presented. These topics include single-pole fo rmulas for simple and higher-order PVs, simple and higher-order poles in double integrals, and products of simple poles in general multiple integrals. Two generalizations of the famous Poincare-Bertrand (PB) th eorem are studied. We then review the following topics: dispersion rel ations for the advanced, retarded, and causal Green's functions; Titch marsh's theorem; applications of the PB theorem to two- and three-part icle loop integrals; and the R and T matrix formalism. Also, various a pplications of the PV methods to nuclear physics, transport theory, an d condensed matter physics are studied. In the appendices several meth ods for evaluating PV integrals, including the Haftel-Tabakin procedur e for calculating the R and T matrices, are reviewed.