Reply to Paris's Comments on Exactification of Stirling's Approximation for the Logarithm of the Gamma Function
| Autor: | Kowalenko, Victor |
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| Rok vydání: | 2014 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: This is an updated version of arXiv:1406.1320. It removes superfluous material from the first version and is to be regarded as the journal version of the original paper |
| Databáze: | arXiv |
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