The method of residues in the evaluation of finite part of algebraically and logarithmically divergent integrals at infinity

Abstract

Three things are done in this paper: (1) We have proposed a theorem that Hadamard Regularization, or the Finite Part Integral, can be extended to divergences at infinity. (2) We have shown that the method of residues neatly extracts the finite parts of an integral, including a non-trivial finite contribution from a great circle contour in the complex plane. (3) Lastly, we applied the theorem we have proposed to quantum field theory, particularly to the scalar quartic theory, where we calculated a one-loop integral that has ultraviolet divergence. It is found that the result from the Finite Part is equal with that from dimensional regularization, for a particular choice of renormalization prescription. Unlike other regularization schemes, the Finite Part already assumes a certain renormalization prescription.