The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation

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The Stokes phenomenon refers to the emergence of jump discontinuities in asymptotic expansions at specific rays in the complex plane. This book presents a radical theory for the phenomenon by ...
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Mellin-Barnes Regularisation

Pp. 82-113 (32)

Victor Kowalenko

Abstract

Ch. 7 presents the theory behind an alternative method of regularising a divergent series known as Mellin-Barnes (MB) regularisation. As a result, the regularised values for more general versions of the two types of terminants presented earlier are derived in terms of MB inte-grals, which are often more expedient to evaluate than the regularised values obtained via Borel summation. Furthermore, unlike the Borel-summed forms for the regularised values, the MB-regularised forms are not affected by Stokes lines and sectors, but are instead valid over domains of convergence, which extend further than Stokes sectors and overlap one another. Thus, there are two different MB-regularised forms for obtaining the regularised value in the common regions of overlapping domains of convergence, which include the Stokes lines of Borel summation. To demonstrate that MB regularisation need not only be applied to an asymptotic series, a numeri-cal example determining the regularised value of an abbreviated version of the binomial theorem is also presented for two different values of the index ρand for various values of the variablez outside the unit disk of absolute convergence.

Affiliation:

Melbourne, Australia