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Exactification is the process of obtaining exact values of a function fromits complete asymptotic expansion. Here Stirling's approximation for thelogarithm of the gamma function or $\ln \Gamma(z)$ is derived completelywhereby it is composed of the standard leading terms and an asymptotic seriesthat is generally truncated. Nevertheless, to obtain values of $\ln \Gamma(z)$,the remainder must undergo regularization. Two regularization techniques arethen applied: Borel summation and Mellin-Barnes (MB) regularization. TheBorel-summed remainder possesses an infinite convergent sum of exponentialintegrals and discontinuous logarithmic terms across Stokes sectors and lines,while the MB-regularized remainders possess one MB integral, which is better tocompute, and similar logarithmic terms. The MB integrals are valid overoverlapping domains of convergence. Hence, two MB-regularized asymptotic formscan be used to evaluate $\ln \Gamma(z)$. Despite having to truncate theBorel-summed remainder, it is found that all the remainders combined with (1)the truncated asymptotic series, (2) the leading terms of Stirling'sapproximation and (3) their logarithmic terms yield identical values of $\ln\Gamma(z)$. In the few cases where the accuracy falls away, it is mostly due toa very high value for the truncation parameter, which results in thecancellation of redundant decimal places. Where possible, all the asymptoticforms yield the same values as the LogGamma routine in Mathematica.

Exactification of Stirling's Approximation for the Logarithm of the Gamma Function