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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Borel Summation of Generalised Termi-nants

Pp. 154-192 (39)

Victor Kowalenko

Abstract

In this chapter general Borel-summed forms for the regularised values of the two types of generalised terminants introduced in the previous chapter are derived for the entire com-plex plane. This is done by expressing both asymptotic series in terms of Cauchy integrals and analysing the singular behaviour as the variablez moves across Stokes sectors. For both types of generalised terminants the Stokes lines represent the complex branches of the singularities in the Cauchy integrals, the difference being that the singularity in the Type I case occurs at −z −β , while for the Type II case it occurs atz −β . Consequently, for a Type I generalised terminant the Cauchy integral represents the regularised value over a primary Stokes sector, whereas for the Type II case, it is the regularised value for a primary Stokes line provided the Cauchy principal value is evaluated. For the other Stokes sectors and lines, the regularised values acquire extra contributions due to the residues of the Cauchy integrals, which emerge each timez −β undergoes a complete revolution. In the case of the Type II generalised terminant, it also acquires an equal and opposite semi-residue contribution oncez −β moves off the primary Stokes line in either di-rection. Hence, the results for the regularised values of both types of generalised terminants are treated separately depending upon whether the singularity undergoes clockwise or anti-clockwise rotations continuously. By referring to the special cases of pstudied in the previous chapter, we find that the Borel-summed forms for the regularised values seldom conform to the conventional view of the Stokes phenomenon. The chapter concludes with the numerical evaluation of the Borel-summed forms for the regularised value of the same Type II generalised terminant at the end of Ch. 9. Though there are more Borel-summed forms to evaluate, in all cases the regularised value obtained from the Borel-summed forms agrees with that obtained from the corresponding MB-regularised forms.

Affiliation:

Melbourne, Australia