the function when $|{\rm arg}\,z| > \pi/n$, since the routine for $\ln \Gamma(z)$ (LogGamma in Mathematica) is restricted to the argument lying in the principal branch of complex plane or to $-\pi <{\rm arg}\,z \leq \pi$. When $z$ is replaced by $z^n$, the routine will only provide values of $\ln \Gamma(z^n)$ for $-\pi/n< {\rm arg}\,z \leq \pi/n$. However, the aim is to determine $f(z)$ over the entire the principal branch. We can use the Borel-summed asymptotic forms for the higher or lower Stokes sectors in this situation, but we also need to be able to verify the results. This can be accomplished by using the MB-regularized asymptotic forms, which represent a different method of evaluating $f(z)$.

Thm. 4.1 presents the MB-regularized asymptotic forms for $\ln \Gamma(z)$, where the regularized value for the remainder of the asymptotic series in the Stirling approximation is expressed in terms of MB integrals with domains of convergence given by $(\pm M-1)\pi \!<\! {\rm arg}\,z \!<\! (\pm M+1)\pi$ and $M$, a non-negative integer. Although there are no Stokes lines and sectors in MB regularization, there is an extra term to the MB-regularized remainder arising from the summation of the residues of the singularities on the Stokes 