The application of the calculus of divergent integrals to Fourier transforms of exponential type functions

Abstract

We consider evaluating the Fourier transform of products of entire and exponential type functions by expressing the Fourier integral in terms of a finite sum of divergent integrals and assigning the divergent integrals values equal to their lower boundary values (LBVs). The method leads to the correct Fourier transform. Moreover, we show from the obtained Fourier transforms that the type of the product of the entire and exponential type functions is equal to the sum of their individual types. Finally, we demonstrate the application of this work in the Fourier integrals arising in environment-free decoherence model.