expressed via Cauchy’s residue theorem in terms of Mellin-Barnes integrals and divergent arc-contour integrals. In the process of regularization the latter integrals are discarded, while the Mellin-Barnes integrals yield finite values, again much like the Hadamard finite part of a divergent integral. Amazingly, the finite values obtained when this technique is applied to an asymptotic expansion of a function yield exact values of the original function, but with one major difference compared with Borel summation. Instead of having to deal with Stokes sectors and lines, we now have to contend with the domains of convergence for the Mellin-Barnes integrals, which not only encompass the former, but also overlap each other. So, while Borel summation and Mellin-Barnes regularization represent techniques for regularizing asymptotic series and yield the same values for the original function from which the complete asymptotic expansion has been derived, they are nevertheless completely different. Moreover, they can be used as a check on one another, which will occur throughout this work.

In the two cases of exactification mentioned above only positive real values of the power variable in the asymptotic expansions were considered, although it was stated that complex values would be studied in the future. As discussed 