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.
