variable to a power. E.g., the asymptotic series for the error function, ${\rm erf}(z)$, which is studied extensively in Ch. 1 of Ref. [@din73], is composed of an infinite series in powers of $1/z^2$. If one wishes to determine values of ${\rm erf}(z)$ over the principal branch via its asymptotic forms, then one requires the asymptotic forms for higher Stokes sectors, not just the asymptotic form for the lowest Stokes sector given by $|{\rm arg}\,z | < \pi/2$. This issue will be discussed in detail later, particularly in Secs. 3 and 5.

The developments in Ref. [@kow09] were primarily concerned with the regularization of the two types of generalized terminants for all Stokes sectors and lines. The term terminant was introduced by Dingle [@din73] after he found that the late terms of the asymptotic series for a host of functions in mathematical physics could be approximated by them. In the present work the aim is to continue with the development of a general theory in asymptotics by using the results in Ref.  [@kow09] as a base. It should be borne in mind that since complete asymptotic expansions are composed of divergent series, we are essentially talking about the development of 