Reply to Paris's Comments on Exactification of Stirling's Approximation for the Logarithm of the Gamma Function

TL;DR

This paper defends the author's previous work on the exactification of Stirling's approximation for the log gamma function, clarifying misconceptions about regularisation, the Stokes phenomenon, and computational methods.

Contribution

It demonstrates the correct application of regularisation, defends the Stokes multiplier definition, and discusses computational advantages of MB-regularised values over Borel-summed forms.

Findings

Confirmed the discontinuous nature of the Stokes phenomenon.

Validated the accuracy of the log gamma function values to 30 decimal places.

Clarified the role of regularisation in asymptotic expansions.

Abstract

In a recent paper [arXiv:1406.1320] Paris has made several comments concerning the author's recent work on the exactification of Stirling's approximation for the logarithm of the gamma function, $\ln \Gamma(z)$. Despite acknowledging that the calculations in Ref. 2 are basically correct, he claims that there is no need for the concept of regularisation when determining values of $\ln \Gamma(z)$ from its complete asymptotic expansion. Here it is shown that he has already applied the concept at the beginning of the analysis in Ref. 1. Next he claims that the definition used in Ref. 2 for the Stokes multiplier in the subdominant part of a complete expansion, which is responsible for demonstrating that the Stokes phenomenon is discontinuous rather than a smooth transition, is not correct. It is shown that the Stokes multiplier used in Ref. 2 is entirely consistent with Berry's description of the conventional view of the Stokes phenomenon in his famous work where he developed the smoothing view [15]. Furthermore, it is pointed out that Paris has not substantiated the smoothing view by at least reproducing or improving the results in Table 7 of Ref. 2, which not only confirm the jump discontinuous nature of the Stokes phenomenon, but also provide accurate values of $\ln \Gamma(z)$ to 30 figures, regardless of the size of the variable or whether the truncation is optimal or not. Finally, the issue of computational expediency of MB-regularised values over Borel-summed values is discussed since Paris is critical of the Borel-summed forms being used in computations.