This introduction outlines the main issues that are to be analysed extensively in the following chapters. It begins with a general description of the Stokes phenomenon in asymptotics. Despite being discovered in 1857, it is pointed out that a complete theory giving the exact size of the jump discontinuities is still not available. Before such a theory can be realised, however, the developments up to the present day need to be discussed. After describing Stokes’s seminal work, the introduction proceeds to the work of Heading and Dingle, a century later. Out of the ob-servations of these three figures a conventional view of the Stokes phenomenon evolved whereby the multipliers of subdominant asymptotic series experience jumps of unity across Stokes lines. Then in 1989 Berry argued that the multipliers experience a rapid smoothing at Stokes lines, al-though there has never been a definitive demonstration of this behaviour. Such a demonstration can only be performed if methods exist for obtaining meaningful values from divergent series, a process referred to here as regularisation. The introduction concludes by discussing the two meth-ods for regularising a divergent series used throughout this book, namely Borel summation and Mellin-Barnes regularisation.

Ch. 2 reviews the derivation of the complete asymptotic expansion for a related func-tion of the error function, which was the first example ever of the Stokes phenomenon. It is shown that the asymptotic method used to derive the expansion, namely the well-known method of iter-ation that is applied to differential equations, introduces an infinity into the remainder and hence, is improper. As a consequence, the concept of an equivalence statement is introduced, while reg-ularisation is defined as the removal of the infinity in an asymptotic series in order to make its remainder summable. The analysis continues by demonstrating that the asymptotic expansion for the variant of the error function changes form at special rays of discontinuity and across specific sectors of the complex plane, which are now called Stokes lines and sectors respectively.

Dingle’s rules for deriving the changing forms of an asymptotic expansion as a result of the Stokes phenomenon are introduced. Of these rules only Nos. 1, 7 and 8 are necessary for the analysis in later chapters. Then with the aid of these rules his theory of terminants is presented, which is important because the asymptotic expansions of a multitude of mathematical functions can be approximated for large values of the summation index by these divergent series whose coefficients possess gamma function growth. There are basically two types of terminants, each possessing different properties according to Dingle’s rules. Meaningful values for both types of terminants are obtained by Borel summation, which is shown to be a method of regularising asymptotic series.

Ch. 4 considers the regularisation of some basic divergent series. The first is the geometric series, which is shown to be conditionally convergent outside the unit circle of absolute convergence forℜz<1 and divergent elsewhere. Nevertheless, the regularised value is found to be identical to the limit of 1/(1−z)when the series is convergent. Then the regularisation of the binomial series is considered, where again, the regularised value is found to equal the limit when the series is convergent. Next the series denoted by 2F1 (a+1,b+1;a+b+2−x; 1), which is divergent forℜx>0, is analysed. Here the regularised value is found to be different from the limit when the series is convergent or for ℜx<0. Finally, the harmonic series is studied, whose regularised value equals Euler’s constant. Unlike the previous examples, the last example involves logarithmic regularisation.

In this chapter the asymptotic forms for the error function and the related function u(a) over the principal branch of the complex plane are regularised via Borel summation. The resulting equations are expressed in terms of a Stokes multiplier, which toggles between -1/2 and 1/2 for the different Stokes sectors. Numerical studies are then conducted for large and small values of the magnitude of the variable, viz. |z|, over the entire principal branch. For the large values of |z| the truncated series is the dominant contribution which is consistent with standard asymptotics. Although the truncated series dominates for small values of|z|, so does the regularised value of its remainder in the opposite sense. Hence, when both contributions are combined, the remaining contribution with the Stokes multiplier can become substantial. Nevertheless, in each case where all the contributions are summed, one always obtains the exact values of the error function. Then an expression for the Stokes multiplier is obtained. By carrying out an extensive numerical analysis in the vicinity of the Stokes line along the positive real axis, it is found that irrespective of the value of variable, the Stokes multiplier is discontinuous and not smooth as implied by the leading order term.

This chapter presents the two main contemporary views of the Stokes phenomenon. In the first view known as Stokes smoothing, it is claimed that rather than experiencing discontinuities at specific rays in the complex plane, an asymptotic expansion, when magnified on a suitable scale, is a rapidly smoothed function. Here we show that this fallacious conclusion has arisen because an asymptotic expansion for the Stokes multiplier has been truncated, thereby giving the misleading impression that it is equal to a term involving the error function when it is in fact only the leading term of a complicated expression that needs to be regularised. Based on resurgence analysis, the second view bears very little semblance to the original discovery made by Stokes. In this view the Stokes lines become analytic curves that are determined by setting the real part of the action in the one-dimensional Schr¨ odinger equation to zero due to a strange interpretation of maximal dominance in a complete asymptotic expansion. Hence, the resulting asymptotic forms are no longer uniform over specific sectors of the complex plane in marked contrast to the conventional view of the Stokes phenomenon.

Ch. 7 presents the theory behind an alternative method of regularising a divergent series known as Mellin-Barnes (MB) regularisation. As a result, the regularised values for more general versions of the two types of terminants presented earlier are derived in terms of MB inte-grals, which are often more expedient to evaluate than the regularised values obtained via Borel summation. Furthermore, unlike the Borel-summed forms for the regularised values, the MB-regularised forms are not affected by Stokes lines and sectors, but are instead valid over domains of convergence, which extend further than Stokes sectors and overlap one another. Thus, there are two different MB-regularised forms for obtaining the regularised value in the common regions of overlapping domains of convergence, which include the Stokes lines of Borel summation. To demonstrate that MB regularisation need not only be applied to an asymptotic series, a numeri-cal example determining the regularised value of an abbreviated version of the binomial theorem is also presented for two different values of the index ρand for various values of the variablez outside the unit disk of absolute convergence.

This chapter is concerned with the numerics of MB-regularised forms for the regu-larised value of a divergent series using the Mathematica software package. To accomplish this, the remainders in the asymptotic forms for u(a)given in Ch. 2 are first MB-regularised. Then an explanation of how to evaluate the resulting MB integrals follows. It is shown that when the values for the MB integrals are added to the truncated asymptotic series and the appropriate Stokes dis-continuous terms, they yield exact values ofu(a)over the principal branch. Because the domains of convergence for the MB integrals extend beyond the Stokes sectors, the two different forms for the regularised value also give exact values of u(a)over their common region of(−π/4,π/4). A similar analysis is then undertaken for the error function erf(z), whose Stokes sectors are shifted compared with those foru(a). A major problem arises when Mathematica attempts to evaluate the MB-regularised values for |argz|>π/2 because the factor ofz −2s in the MB integrals lies outside the principal branch. However, with the introduction of the seemingly innocuous factor of exp(2πi jk)into the asymptotic series, different MB-regularised forms are obtained with domains of convergence that encompass the previously inaccessible sectors of the principal branch. Con-sequently, the Stokes multiplier equals -1/2 for j=±1, while it equals 1/2 forj=0 as in Ch. 5. When a numerical analysis is undertaken for|argz|>π/2 with the new forms for the regularised value, exact values of the error function are obtained irrespective of the magnitude ofz.

Generalised terminants are produced when the coefficients of the two types of series considered in Ch. 7 are set equal toΓ(pk+q), where pandqare both real and positive and the variablez is altered to z β , where βcan be much greater than unity. Ch. 9 is concerned with the derivation of the MB-regularised forms for the regularised values of both types of generalised terminants over the entire complex plane, which are presented in Propositions 4 and 5. These results are then simplified by considering special cases of p, the first where it is the reciprocal of a natural natural number and the second, where it equals 2. The chapter concludes by evaluating the regularised value of a Type II generalised terminant withβ=6,p=1 andq=1/5 using the various MB-regularised forms that apply over the principal branch forz. Because there is no known special function equivalent for this asymptotic series, the results from this study serve as a test-bed for the results in the following chapter. Nevertheless, it is found that the regularised values obtained from the two MB-regularised forms for each of the six common regions of the overlapping domains of convergence equal one another for both small and large values of|z|.

In this chapter general Borel-summed forms for the regularised values of the two types of generalised terminants introduced in the previous chapter are derived for the entire com-plex plane. This is done by expressing both asymptotic series in terms of Cauchy integrals and analysing the singular behaviour as the variablez moves across Stokes sectors. For both types of generalised terminants the Stokes lines represent the complex branches of the singularities in the Cauchy integrals, the difference being that the singularity in the Type I case occurs at −z −β , while for the Type II case it occurs atz −β . Consequently, for a Type I generalised terminant the Cauchy integral represents the regularised value over a primary Stokes sector, whereas for the Type II case, it is the regularised value for a primary Stokes line provided the Cauchy principal value is evaluated. For the other Stokes sectors and lines, the regularised values acquire extra contributions due to the residues of the Cauchy integrals, which emerge each timez −β undergoes a complete revolution. In the case of the Type II generalised terminant, it also acquires an equal and opposite semi-residue contribution oncez −β moves off the primary Stokes line in either di-rection. Hence, the results for the regularised values of both types of generalised terminants are treated separately depending upon whether the singularity undergoes clockwise or anti-clockwise rotations continuously. By referring to the special cases of pstudied in the previous chapter, we find that the Borel-summed forms for the regularised values seldom conform to the conventional view of the Stokes phenomenon. The chapter concludes with the numerical evaluation of the Borel-summed forms for the regularised value of the same Type II generalised terminant at the end of Ch. 9. Though there are more Borel-summed forms to evaluate, in all cases the regularised value obtained from the Borel-summed forms agrees with that obtained from the corresponding MB-regularised forms.

In order to demonstrate that it is regularisation and not Borel summation which is responsible for yielding meaningful values to asymptotic series, the gamma function in both types of generalised terminants is now replaced by the functionf(pk+q)in this chapter. Then the regu-larised values for both types of series are determined by assuming that f(s)is a Mellin transform. These appear in Propositions 6 and 7. Although both types of series are different from generalised terminants, the proofs of the propositions are nonetheless based on the exposition in the preceding chapter. Consequently, the regularised values are referred to as extended Borel-summed forms. The chapter concludes by considering a complicated example of a Type II series, where the coef-ficients can be expressed as the Mellin transform of the product of the Bessel functionJ ν (x)and the Macdonald function Kν (x). As there is no special function equivalent to this asymptotic series, the MB-regularised forms for the regularised value are derived with the aid of the general theory in Ch. 7. Then a numerical study of both the extended Borel-summed and MB-regularised forms is carried out with the index νset equal to 1/3 and -3/5 and for large and small values of|z| over the principal branch. Once again, the Borel-summed and MB-regularised forms yield identical regularised values for the series.

In the final chapter a summary of the major issues surrounding this work is presented together with a discussion of the main results. The conclusion begins by relating the discovery of the Stokes phenomenon in the terms of the concepts appearing in this book plus the subse-quent developments over the next sesquicentenary or so. It then proceeds to explain why these developments have been unsuccessful in providing a satisfactory explanation of the phenomenon. This is attributed to the fact that they have not employed the important concepts of equivalence and regularisation, which are crucial for obtaining meaningful values from each component se-ries in a complete asymptotic expansion. Such values have been referred to as regularised values throughout this work. Then the two main techniques of regularising a divergent series, viz. Borel summation and MB-regularisation, are discussed and compared with each other. It is pointed out that the former is responsible for the behaviour of an asymptotic expansion as it experiences the Stokes phenomenon, while the latter produces broader sectors over which an asymptotic ex-pansion is valid. Not only do the broader sectors overlap one another, there is no evidence of discontinuities across specific rays as there is with Borel-summed forms for the regularised value of an asymptotic series. Furthermore, the concept of regularisation allows us to go beyond Borel summation and consider asymptotic series whose coefficients are not dependent upon the gamma function. Finally, the ramifications of this work on the subject of asymptotics are addressed.

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