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 formulas 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 Poincaré-Bertrand (PB) theorem are studied. We then review the following topics: dispersion relations for the advanced, retarded, and causal Green's functions; Titchmarsh's theorem; applications of the PB theorem to two- and three-particle loop integrals; and the R and T matrix formalism. Also, various applications of the PV methods to nuclear physics, transport theory, and condensed matter physics are studied. In the appendices several methods for evaluating PV integrals, including the Haftel-Tabakin procedure for calculating the R and T matrices, are reviewed.