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

TL;DR

This paper critically examines the exponentially improved expansion of log gamma function, confirming recent results, correcting a definition of the Stokes multiplier, and demonstrating the smooth Stokes phenomenon transition through numerical example.

Contribution

It shows that Kowalenko's regularisation approach yields an equivalent expansion and corrects his interpretation of the Stokes phenomenon as a jump discontinuity.

Findings

Kowalenko's regularisation produces an equivalent expansion.

The Stokes phenomenon is a smooth transition, not a jump.

Numerical example demonstrates the smooth transition of exponential terms.

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 $\arg\,z=\fs\pi$.