astonishing accuracy. This was necessary in order to observe the effect of the subdominant asymptotic series, which required in some instances that the analysis be conducted to 65 decimal places as described in Sec. 7 of Ref.  [@kow95].

In the second case [@kow002] exact values of Bessel and Hankel functions were calculated from their well-known asymptotic expansions given in Ref. [@wat95]. In this instance there were no subdominant exponential series because the analysis was restricted to positive real values of the variable. However, unlike Ref. [@kow95], different values or levels of truncation were applied to the asymptotic series. Whilst the truncated asymptotic series yielded a different value for a fixed value of the variable, when it was added to the corresponding regularized value for its remainder, the actual value of the Bessel or Hankel function was calculated to within the machine precision of the computing system. The level of truncation was governed by an integer parameter $N$, which will also be introduced here. In fact, the truncation parameter as it is called will play a much greater role here since it will be set equal to much larger values than those in Ref. [@kow002].

Because a complete asymptotic expansion is 