more rapidly even for relatively small values of the truncation parameter.

At this stage the Borel-summed asymptotic forms have been shown to yield $\ln \Gamma(z)$ for real values of $z$. That is, exactification of the asymptotic form for a function has been achieved for real values of $z$ as in the previous instances studied in Refs.[@kow95] and [@kow002]. Now, the analysis is extended to complex values of $z$. This is done by putting $z=|z| \exp(i \theta)$, where $|z|$ equals either 3 or 1/10 and $\theta$ ranges over the principal branch of the complex plane, i.e. $-\pi< \theta \leq \pi$. In addition, the regularized value of the remainder is evaluated by using the Borel-summed forms in Thm. 2.1. That is, instead of relying on a routine that can calculate values of the incomplete gamma function, we evaluate the exponential integrals appearing in Eqs.(\[sixty\]) and (\[sixtya\]) by using the NIntegrate routine in Mathematica, which has the effect of increasing the time of computation substantially. For $|z|=3$ and $N$ below the optimal point of truncation, it is found that the leading terms of the Stirling’s approximation are accurate to the first couple of decimal places to $\ln \Gamma(z)$, but not the thirty figure 