Exactification of Stirling’s Approximation for the Logarithm of the Gamma Function Victor Kowalenko School of Mathematics and Statistics The University of Melbourne Victoria 3010, Australia. February 16, 2021 Abstract Exactification is the process of obtaining exact values of a function from its complete asymptotic expansion. This work studies the complete form of Stirling’s approximation for the logarithm of the gamma function, which consists of standard leading terms plus a remainder term involving an infinite asymptotic series. To obtain values of the function, the divergent remainder must be regularized. Two regularization techniques are introduced: Borel summation and Mellin-Barnes (MB) regularization. The Borel-summed remainder is found to be composed of an infinite convergent sum of exponential integrals and discontinuous logarithmic terms from crossing Stokes sectors and lines, while the MB-regularized remainders possess one MB integral, with similar logarithmic terms. Because MB integrals are valid over overlapping domains of convergence, two MB-regularized asymptotic forms can often be used to evaluate the logarithm of the gamma function. Although the Borel- summed remainder is truncated, albeit at very large values of the sum, it is found that all the remainders when combined with (1) the truncated asymptotic series, (2) the leading terms of Stirling’s approximation and (3) their logarithmic terms yield identical valuesDRAFT that agree with the very high precision results obtained from mathematical software packages.February 16, 2021-- 02:14 arXiv:1404.2705v3 [math.CA] 13 Feb 2021 Keywords: Asymptotic series, Asymptotic form, Borel summation, Complete asymp- totic expansion, Discontinuity, Divergent series, Domain of convergence, Exactification, Gamma function, Mellin-Barnes regularization, Regularization, Remainder, Stokes dis- continuity, Stokes line, Stokes phenomenon, Stokes sector, Stirling’s approximation 2010 Mathematics Subject Classification: 30B10, 30B30, 30E15, 30E20, 34E05, 34E15, 40A05, 40G10, 40G99, 41A60 email: [email protected] 1 1 Introduction In asymptotics exactification is defined as the process of obtaining the exact values of a function/integral from its complete asymptotic expansion and has already been achieved in two notable cases. For those unfamiliar with the concept, a complete asymptotic ex- pansion is defined as a power series expansion for a function or integral that not only possesses all the terms in a dominant asymptotic series, but also all the terms in fre- quently neglected subdominant or transcendental asymptotic series, should they exist. The latter series are said to lie beyond all orders, while the methods and theory behind them belong to the discipline or field now known as asymptotics beyond all orders or exponential asymptotics [1]. One outcome of this relatively new field is that it seeks to obtain far more accurate values from the asymptotic expansions for functions/integrals than standard Poincare´easymptotics [2]. These calculations, which often yield values that are accurate to more than twenty decimal places, are referred to as hyperasymptotic evaluations or hyperasymptotics, for short. Hence exactification represents the extreme of hyperasymptotics. In the first successful case of exactification exact values of a particular case of the 3 generalized Euler-Jacobi series, viz. S (a) = ∞ exp( an ), were evaluated from its 3 n=1 − the complete asymptotic expansion, which was given in powers of a. Although it had been found earlier in Ref. [3] that there couldP be more than one subdominant series in the complete asymptotic expansion for the generalized Euler-Jacobi series, the complete asymptotic expansion for S3(a) was found to be composed of an infinite dominant algebraic series and another infinite exponentially-decaying asymptotic series, whose coefficients resembled those appearing in the asymptotic series for the Airy function Ai(z). In carrying out the exactification of this complete asymptotic expansion, a range of values for a was considered with the calculations performed to astonishing accuracy. This was necessary in order to observe the effect of the subdominant asymptotic series, which required in some instances that the analysis be conducted to 65 decimal places as described in Sec. 7 of Ref. [3]. In the second case [4] exact values of Bessel and Hankel functions were calculated from their well-known asymptotic expansions given in Ref. [5]. In this instance there were no subdominant exponential series because the analysis was restricted to positive real values of the variable. However, unlike Ref. [3], different values or levels of truncation were applied to the asymptotic series. Whilst the truncated asymptotic series yielded a different value for a fixed value of the variable, when it was added to the corresponding regularized value for its remainder, the actual value of the Bessel or Hankel function was calculated to within the machine precision of the computing system. The level of truncation was governed by an integer parameter N, which will also be introduced here. In fact, the truncation parameter as it is called will play a much greater role here since it will be set equal to much larger values than those in Ref. [4]. Because a complete asymptotic expansion is composed of divergent series, exactifica- tion involves being able to obtain meaningful values from such series. To evaluate these, one must introduce the concept of regularization, which is defined in this work as the removal of the infinity in the remainder of an asymptotic series in order to make the series summable. It was first demonstrated in Ref. [6] that the infinity appearing in the remainder of an asymptotic series arises from an impropriety in the method used to derive it. Consequently, regularization was seen as a necessary means of correcting an asymp- 2 totic method such as the method of steepest descent or iteration of a differential equation. Regularization was also shown to be analogous to taking the finite or Hadamard part of a divergent integral [6]- [9]. Two very different techniques will be used to regularize the divergent series appearing throughout this work. As described in Refs. [4, 7], the most common method of regular- izing a divergent series is Borel summation, but often, it produces results that are not amenable to fast and accurate computation. To overcome this drawback, the numerical technique of Mellin-Barnes regularization was developed for the first time in Ref. [3]. In this regularization technique divergent series are expressed via Cauchy’s residue theorem in terms of Mellin-Barnes integrals and divergent arc-contour integrals. In the process of regularization the latter integrals are discarded, while the Mellin-Barnes integrals yield finite values, again much like the Hadamard finite part of a divergent integral. Amazingly, the finite values obtained when this technique is applied to an asymptotic expansion of a function yield exact values of the original function, but with one major difference com- pared with Borel summation. Instead of having to deal with Stokes sectors and lines, we now have to contend with the domains of convergence for the Mellin-Barnes integrals, which not only encompass the former, but also overlap each other. So, while Borel sum- mation and Mellin-Barnes regularization represent techniques for regularizing asymptotic series and yield the same values for the original function from which the complete asymp- totic expansion has been derived, they are nevertheless completely different. Moreover, they can be used as a check on one another, which will occur throughout this work. In the two cases of exactification mentioned above only positive real values of the power variable in the asymptotic expansions were considered, although it was stated that complex values would be studied in the future. As discussed in the preface to Ref. [3], such an undertaking represents a formidable challenge because as the variable in an asymptotic series moves about the complex plane or its argument changes, a complete asymptotic expansion experiences significant modification due to the Stokes phenomenon [10]. This means that at particular rays or lines in the complex plane, an asymptotic expansion develops jump discontinuities, which can result in the emergence of an extra asymptotic series in the complete asymptotic expansion. Thus, a complete asymptotic expansion is only uniform over either a sector or a ray in the complex plane, which means in turn that in order to exactify a complete asymptotic expansion over all arguments or phases of the variable in the power series, one requires a deep understanding of the Stokes phenomenon. This understanding entails: (1) being able to determine the locations of all jump discontinuities, and (2) solving the more intricate problem of their quantification when they do occur. Because of the Stokes phenomenon, it becomes necessary not only to specify a com- plete asymptotic expansion, but also the range of the argument of the variable in the power series of the expansion. The combination of two such statements are referred to as asymptotic forms in this work. In particular, it should be noted that if the same complete asymptotic expansion is valid for different sectors or rays of the complex plane, then in each instance the original function being studied will also be different. A major advance in enabling asymptotic forms to be evaluated over all values of the argument of the power series variable occurred with the publication of Ref. [7], which began