Comments on "Exactification of Stirling's approximation for the logarithm of the gamma function"

Abstract

We re-examine the exponentially improved expansion for $\log\,\g(z)$, first considered in Paris and Wood in 1991, to point out that the recent treatment by Kowalenko [Exactification of Stirling's approximation for the logarithm of the gamma function, arXiv:1404.2705] using his procedure of regularisation produces an equivalent result. In addition, we point out an error in his definition of the Stokes multiplier that leads him to make the incorrect statement that the Stokes phenomenon is a jump discontinuity, rather than a smooth transition. We supply a numerical example that clearly demonstrates the smooth transition of the leading subdominant exponential $e^{2\pi iz}$ across the Stokes line $\arg\,z=\fs\pi$.