$\Gamma(z)$ from the complex numerical values calculated for $\ln \Gamma(z)$. Despite this, however, no one has ever been able to render exact values of $\Gamma(z)$ or $\ln \Gamma(z)$ when either function has been expressed in terms of the Stirling approximation because when all the terms in the approximation are considered, it becomes an asymptotic expansion. Consequently, it is not absolutely convergent. Here we aim to investigate how the results in Ref. [@kow09] can be used to determine exact values of the complete version of Stirling’s approximation of $\ln \Gamma(z)$ for all values of ${\rm arg}\, z$. In short, this paper aims to exactify Stirling’s approximation.

Sec. 2 introduces the standard form of Stirling’s approximation for $\ln \Gamma(z)$ as given in Ref. [@abr65]. Then $\ln \Gamma(z)$ is expressed in terms of the specific leading order terms associated with the approximation and a truncated asymptotic power series, whose coefficients are related to the Bernoulli numbers. Although it can be divergent, the latter series is often neglected in accordance with standard Poincar$\acute{\rm e}$ asymptotics [@whi73]. Moreover, it is not known whether inclusion of the asymptotic series represents a complete asymptotic expansion for $\ln \Gamma(z)$. Consequently, we turn to Binet’s second expression of $\ln 