The Application of Zeta Regularization Method to the Calculation of Certain Divergent Series and Integrals

Abstract

In this paper, we generalize the Zeta regularization method to the divergent integrals for positive ‘s.’ Using the Euler-Maclaurin summation formula we express a divergent integral in terms of a linear combination of divergent series, which can be regularized using the Riemann Zeta function, s>0. For the case of the pole at s=1, we use a property of the functional determinant to obtain the regularization. With the aid of the Laurent series, we extend the Zeta regularization to the case of integral. We believe that this method can be of interest in the regularization of the divergent UV integrals in quantum field theory since it does not have the problems of the analytic regularization or dimensional regularization.