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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Numerics of MB-Regularised Forms

Pp. 114-132 (19)

Victor Kowalenko

Abstract

This chapter is concerned with the numerics of MB-regularised forms for the regu-larised value of a divergent series using the Mathematica software package. To accomplish this, the remainders in the asymptotic forms for u(a)given in Ch. 2 are first MB-regularised. Then an explanation of how to evaluate the resulting MB integrals follows. It is shown that when the values for the MB integrals are added to the truncated asymptotic series and the appropriate Stokes dis-continuous terms, they yield exact values ofu(a)over the principal branch. Because the domains of convergence for the MB integrals extend beyond the Stokes sectors, the two different forms for the regularised value also give exact values of u(a)over their common region of(−π/4,π/4). A similar analysis is then undertaken for the error function erf(z), whose Stokes sectors are shifted compared with those foru(a). A major problem arises when Mathematica attempts to evaluate the MB-regularised values for |argz|>π/2 because the factor ofz −2s in the MB integrals lies outside the principal branch. However, with the introduction of the seemingly innocuous factor of exp(2πi jk)into the asymptotic series, different MB-regularised forms are obtained with domains of convergence that encompass the previously inaccessible sectors of the principal branch. Con-sequently, the Stokes multiplier equals -1/2 for j=±1, while it equals 1/2 forj=0 as in Ch. 5. When a numerical analysis is undertaken for|argz|>π/2 with the new forms for the regularised value, exact values of the error function are obtained irrespective of the magnitude ofz.

Affiliation:

Melbourne, Australia