Divergent series:taming the tails

M. V. Berry1 & C. J. Howls2

1 H. H. Wills Physics Laboratory, TyndallAvenue, Bristol BS8 1TL, UK2 Mathematical Sciences, University ofSouthampton, Southampton, SO17 1BJ, UK.

1 Introduction

By the 17th century, in what became the theoryof convergent series, it was beginning to be under-stood how a sum of infinitely many terms couldbe finite; this is now a fully developed and largelystandard element of every mathematician’s edu-cation. Contrasting with it is the theory of seriesthat do not converge, especially those in whichthe terms first get smaller but then increase fac-torially: this is the class of ‘asymptotic series’, en-countered frequently in applications, with whichthis article is mainly concerned. Although now avibrant area of research, the development of thetheory of divergent series has been tortuous andoften accompanied by controversy. As a peda-gogical device to explain the subtle concepts in-volved, we will focus on the contributions of indi-viduals and describe how the ideas developed dur-ing several (overlapping) historical epochs, oftendriven by applications ranging from wave physicsto number theory. This article complements theaccompanying Companion article by P D Milleron ‘Perturbation Theory (including AsymptoticExpansions)’.

2 The Classical Period

In 1747, the Reverend Thomas Bayes (betterknown for his theorem in probability theory)sent a letter to Mr John Canton, F.R.S; it waspublished posthumously in 1763. Bayes demon-strated that the series now known as Stirling’sexpansion for log(z!), “asserted by some eminentmathematicians,” does not converge. Arguingfrom the recurrence relation relating successiveterms of the series, he showed that the coefficients“increase at a greater rate than what can be com-pensated by an increase of the powers of z, thoughz represent a number ever so large.” As we would

say now, this expansion of the factorial functionis a factorially divergent asymptotic series. Theexplicit form of the series, written formally as anequality, is

log(z!) = (z + 1/2) log z + log√

2π − z

+1

2π2z

∞∑r=0

(−1)rar

(2πz)2r,

where

ar = (2r)!

∞∑n=1

1

n2r+2.

Bayes claimed that Stirling’s series “can neverproperly express any quantity at all” and themethods used to obtain it “are not to be dependedupon.”

Leonhard Euler, in extensive investigations ofa wide variety of divergent series beginning sev-eral years after Bayes, took the opposite view.He argued that such series have a precise mean-ing, to be decoded by suitable resummation tech-niques (several of which he invented): “if we em-ploy [the] definition . . . that . . . the sum of a se-ries is that quantity which generates the series,all doubts with respect to divergent series vanishand no further controversy remains.”

With the development of rigorous analysis inthe 19th century, Euler’s view, which as we willsee is the modern one, was sidelined and evenderided. As Niels Henrik Abel wrote in 1828:“Divergent series are the invention of the devil,and it is shameful to base on them any demon-stration whatsoever.” Nevertheless, divergent se-ries, especially factorially divergent ones, repeat-edly arose in application. Towards the end of thecentury they were embraced by Oliver Heaviside,who used them in pioneering studies of radio wavepropagation. He obtained reliable results usingundisciplined semi-empirical arguments that werecriticised by mathematicians, much to his disap-pointment: “It is not easy to get up any enthu-siasm after it has been artificially cooled by thewet blanket of rigorists.”

3 The Neoclassical Period

In 1886, Henri Poincare published a definition ofasymptotic power series, involving a large param-eter z, that was both a culmination of previous

1

2

0 10 20 30 40-8

-6

-4

-2

log 10

|erro

r|

n

zz

Ai(z)

Figure 1: Left: rainbow-crossing variable, along any line transverse to the rainbow curve. Middle: Airy function.

Right: error from truncating asymptotic series for Ai(z) at the term n− 1, for z = 5.24148; optimal truncation

occurs at the nearest integer to F = 4z3/2/3, i.e. n = 16.

work by analysts and the foundation of much ofthe rigorous mathematics that followed. A seriesof the form

∑∞n=0 an/z

n is defined as asymptoticby Poincare if the error resulting from truncationat the term n = N vanishes as K/zN+1(K > 0)as |z| → ∞ in a certain sector of the complex zplane. In retrospect, Poincare’s definition seems aretrograde step, because although it encompassesconvergent as well as divergent series in one the-ory, it fails to address the distinctive features ofdivergent series that ultimately lead to the correctinterpretation that can also cure their divergence.

It was George Stokes, in research inspired byphysics nearly four decades before Poincare, wholaid the foundations of modern asymptotics. Hetackled the problem of approximating an inte-gral devised by George Airy to describe wavesnear caustics, the most familiar example beingthe rainbow. This is what we now call the Airyfunction Ai(z), defined by the oscillatory integral

Ai(z) ≡ 1

2π

∫ ∞−∞

exp

(i

3t3 + izt

)dt,

the rainbow-crossing variable being Rez (Fig-ure 1) and the light intensity being Ai2(z). Stokesderived the asymptotic expansion representingthe Airy function for z > 0, and showed that it isfactorially divergent. His innovation was to trun-cate this series not at a fixed order N but at itssmallest term (optimal truncation), correspond-ing to an order N(z) that increases with z. Bystudying the remainder left after optimal trun-cation, he showed that it is possible to achieveexponential accuracy (Figure 1) far beyond thepower-law accuracy envisaged in Poincare’s defi-

nition. We will call such optimal truncation su-perasymptotics.

Superasymptotics enabled Stokes to under-stand a much deeper phenomenon, one that isfundamental to the understanding of divergent se-ries. In Ai(z), z > 0 corresponds to the dark sideof the rainbow, where the function decays expo-nentially: physically, this represents an evanes-cent wave. On the bright side z < 0, the functionoscillates trigonometrically, that is, as the sum oftwo complex exponential contributions, each rep-resenting a wave; the interference of these wavesgenerates the ‘supernumerary rainbows’ (whoseobservation was one of the phenomena earlier ad-duced by Thomas Young in support of his viewthat light is a wave phenomenon). One of thesecomplex exponentials is the continuation acrossz = 0 of the evanescent wave on the dark side.But where does the other originate?

Stokes’s great discovery was that this secondexponential appears during continuation of Ai(z)in the complex plane from positive to negative z,across what is now called a ‘Stokes line’, wherethe dark-side exponential reaches its maximumsize. Alternatively stated, the small (subdomi-nant) exponential appears when maximally hid-den behind the large (dominant) one. For Ai(z)the Stokes line is arg z = 120◦ (Figure 2).

Stokes thought that the least term in theasymptotic series representing the large exponen-tial constitutes an irreducible vagueness in the de-scription of Ai(z) in his superasymptotic scheme.By quantitative analysis of the size of this leastterm, Stokes concluded that only at maximaldominance could this obscure the small expo-

3

Rez

ImzStokes line

θ

0˚180˚

120˚

Figure 2: Complex plane of argument z (whose real

part is the z of Figure 1) of Ai(z), showing Stokes line

at arg z = 120◦.

nential, which could then appear without incon-sistency. As we will explain later, Stokes waswrong to claim that superasymptotics—optimaltruncation—represents the best approximationthat can be achieved within asymptotics. Buthis identification of the Stokes line with the birthof the small exponential (Figure 3) was correct.Moreover, he also appreciated that the conceptwas not restricted to Ai(z) but applies to a widevariety of functions arising from integrals, so-lutions of differential equations and recurrenceequations, etc., for which the associated asymp-totic series are factorially divergent.

This Stokes phenomenon, connecting differentexponentials representing the same function, iscentral to our current understanding of such di-vergent series, and is the feature that distin-guishes them most sharply from convergent ones.In view of this seminal contribution, it is ironicthat George (‘G H’) Hardy makes no mention ofthe Stokes phenomenon in his textbook ‘Diver-gent series’. Nor does he exempt Stokes from hisdevastating assessment of 19th century Englishmathematics: “there [has been] no first-rate sub-ject, except music, in which England has occupiedso consistently humiliating a position. And whathave been the peculiar characteristics of such En-glish mathematics . . . ? . . . for the most part, am-ateurism, ignorance, incompetence, and trivial-ity.”

0.2 0.4 0.6 0.8

-0.4

-0.2

0 0

a

edcb

ImAi(z)ReAi(z)

Stokesjump

Figure 3: Approximations to Ai(z exp(iθ)) in the Ar-

gand plane, (Re(Ai), Im(Ai)) for z = 1.31 . . ., i.e. F =

4z3/2/3 = 2, plotted parametrically from θ = 0◦ (•)

to θ = 180◦ (�). The curves are: (a) exact Ai; (b)

lowest-order asymptotics (no correction terms); (c)

optimal truncation without Stokes jump; (d) optimal

truncation including Stokes jump; (e) optimal trunca-

tion including smoothed Stokes jump. For this value

of F , the optimally truncated sum contains only two

terms; on the scale shown, the Stokes jump would be

invisible for larger F . Note that without the Stokes

jump (curves b and c) the asymptotics must devi-

ate from the exact function beyond the Stokes line at

θ = 120◦.

4 The Modern Period

Late in the 19th century, Jean-Gaston Darbouxshowed that for a wide class of functions the highderivatives diverge factorially. This would be-come an important ingredient in later research,for the following reason. Asymptotic expan-sions (particularly those encountered in physicsand applied mathematics) are often based on lo-cal approximations: the steepest-descent methodfor approximating integrals is based on local ex-pansion about a saddle-point, the phase-integralmethod for solving differential equations (e.g.the Wentzel-Kramers-Brillouin (WKB) approxi-mation to Schrodinger’s equation in quantum me-chanics) is based on local expansions of the coef-ficients, etc. Therefore successive orders of ap-proximation involve successive derivatives, and

4

the high orders, responsible for the divergence ofthe series, involve high derivatives.

Another major late 19th century ingredient ofour modern understanding was Emile Borel’s de-velopment of a powerful summation method inwhich the factorials causing the high orders to di-verge are tamed by replacing them by their inte-gral representation. Often this enables the seriesto be summed ‘under the integral sign’. Underly-ing the method is the formal equality

∞∑r=0

arzr

=

∞∑r=0

arr!

zrr!=

∫ ∞0

dt e−t∞∑r=0

arr!

(t

z

)r

.

Reading this from right to left is instructive.Interchanging summation and integration showswhy the series on the left diverges if the ar in-crease factorally (as in the cases we are consid-ering): the integral is over a semi-infinite range,yet the sum in the integrand converges only for|t/z| < 1. Borel summation effectively repairs ananalytical transgression that may have caused thedivergence of the series. The power of Borel sum-mation is that, as was fully appreciated only later,it can be analytically continued across Stokeslines, where some other summation techniquesfail (for example Pade approximants).

Now we come to the central development inmodern asymptotics. In a seminal and vision-ary advance, motivated initially by mathemati-cal difficulties in evaluating some integrals occur-ring in solid-state physics and developed in a se-ries of papers culminating in a book publishedin 1973, Robert Dingle synthesized earlier ideasinto a comprehensive theory of factorially diver-gent asymptotic series.

Dingle’s starting point was Euler’s insight thatdivergent series are obtained by a sequence ofprecisely specified mathematical operations onthe integral or differential equation defining thefunction being approximated, so the resulting se-ries must represent the function exactly, albeit incoded form, which it is the task of asymptotics todecode. Next was the realization that Darboux’sexpression of high derivatives in terms of factori-als implies that the high orders of a wide classof asymptotic series diverge similarly. In turnthis means that the terms beyond Stokes’s opti-mal truncation—representing the tails of such se-ries beyond superasymptotics—can all be Borel-

summed in the same way.The next insight was Dingle’s most original

contribution. Consider a function representedby several different formal asymptotic series (forexample those corresponding to the two expo-nentials in Ai(z)), each representing the func-tion differently in sectors of the complex planeseparated by Stokes lines. Since each series isa formally complete representation of the func-tion, each must contain, coded into its high or-ders, information about all the other series. ThusDarboux’s factorials are simply the first terms ofasymptotic expansions of each of the late termsof the original series. Dingle appreciated thatthe natural variables implied by Darboux’s theoryare the differences between the various exponents;usually these are proportional to the large asymp-totic parameter. In the simplest case, where thereare only two exponentials, there is one such vari-able, which Dingle called the singulant, denotedF . For the Airy function Ai(z), F = 4z3/2/3.

We exhibit Dingle’s expression for the high or-ders for an integral with two saddle-points a andb, corresponding to exponentials exp(−Fa) andexp(−Fb) with Fab = Fb − Fa and series with

terms T(a)r and T

(b)r : for r � 1, the terms of the

a series are related to those of the b series by

T (a)r = K

(r − 1)!

F rab

(T

(b)0 +

Fab

(r − 1)T

(b)1

+F 2ab

(r − 1)(r − 2)T

(b)2 + . . .

),

in which K is a constant. This shows that al-though the early terms T

(a)0 , T

(a)1 , T

(a)2 , . . . of an

asymptotic series can rapidly get extremely com-plicated, the high orders display a miraculousfunctional simplicity.

With Borel’s as the chosen summation method,Dingle’s late terms formula enabled the diver-gent tails of series to be summed in terms of cer-tain terminant integrals, and then re-expandedto generate new asymptotic series, exponentiallysmall compared with the starting series. He envis-aged that “these terminant expansions can them-selves be closed with new terminants; and so on,stage after stage.” Such resummations, beyondsuperasymptotics, were later called hyperasymp-totics.

Thus Dingle envisaged a universal technique

5

for repeated resummation of factorially divergentseries, to obtain successively more accurate ex-ponential improvements far beyond that achiev-able by Stokes’s optimal truncation of the orig-inal series. The meaning of universality is thatalthough the early terms—the ones that get suc-cessively smaller—can be very different for dif-ferent functions, the summation method for thetails is always the same, involving terminant in-tegrals that are the same for a wide variety offunctions. The method automatically incorpo-rates the Stokes phenomenon. Although Dingleclearly envisaged the hyperasymptotic resumma-tion scheme as described above, he applied it onlyto the first stage; this was sufficient to illustratethe high improvement in numerical accuracy ascompared with optimal truncation.

Like Stokes before him, Dingle presented hisnew ideas not in the ‘lemma, theorem, proof’ stylefamiliar to mathematicians, but in the discursivemanner of a theoretical physicist. Perhaps thisis why it has taken several decades for the origi-nality of his approach to be widely appreciatedand accepted. Meanwhile his explicit relation,connecting the early and late terms of differentasymptotic series representing the same function,was rediscovered independently by several people.In particular, Jean Ecalle coined the term resur-gence for the phenomenon, and in a sophisticatedand comprehensive framework applied it to a verywide class of functions.

5 The Postmodern Period

One of the first steps beyond superasymptoticsinto hyperasymptotics was an application of Din-gle’s ideas to give a detailed description of theStokes phenomenon. In 1988, one of us (MichaelBerry) resummed the divergent tail of the dom-inant series of the expansion, near a Stokes line,of a wide class of functions including Ai(z), togive birth to the change in the subdominant ex-ponential, occurring not suddenly as in previousaccounts of the phenomenon, but smoothly andin a universal manner. In terms of Dingle’s sin-gulant F , now defined as the difference betweenthe exponents of the dominant and subdominantexponentials, the Stokes line corresponds to thepositive real axis in the complex F plane, asymp-

0 60 120 1800

0.5

1

mul

tiplie

r

θ˚

F=3

F=10

Stokes line

Figure 4: Stokes multiplier (full curve) and error-

function smoothing (dashed curve), for Ai(z exp(iθ)),

for z = 1.717 . . . (i.e. F = 4z3/2/3 = 3) and z =

3.831 . . . (i.e. F = 4z3/2/3 = 10).

totics corresponds to ReF � 1, and the Stokesphenomenon corresponds to crossing the Stokesline, that is ImF passing through zero.

The result of the resummation is thatthe change in the coefficient of the smallexponential—the Stokes multiplier—is universalfor all factorially divergent series, and propor-tional to

1

2

(1 + Erf

(ImF√2ReF

)).

In the limit ReF → ∞ this becomes the unitstep. For ReF large but finite, the formula de-scribes the smooth change in the multiplier (Fig-ure 4), and makes precise the description givenby Stokes in 1902 after thinking about divergentseries for more than half a century: “. . . the infe-rior term enters as it were into a mist, is hiddenfor a little from view, and comes out with its co-efficient changed. The range during which theinferior term remains in a mist decreases indef-initely as the [large parameter] increases indefi-nitely.” The smoothing shows that the ‘range’referred to by Stokes, that is the effective thick-ness of the Stokes line, is of order

√ReF .

The full hyperasymptotic repeated resumma-tion scheme envisaged by Dingle has been imple-mented in several ways. We (the present authors)investigated one-dimensional integrals with sev-eral saddle-points, each associated with an ex-ponential and its corresponding asymptotic se-

6

-10

-15

-5

0

log 10

|term

| zeroth stage

12 terms

6 terms

3 terms 2

terms1st stage

3rd2nd

Figure 5: Terms in the first four stages in the

hyperasymptotic approximation to Ai(4.326 . . .) =

4.496 . . . × 10−4, i.e. F = 4z3/2/3 = 12, normal-

ized so that the lowest approximation is unity. For

the lowest approximation, i.e. no correction terms,

the fractional error is ε ≈ 0.01; after stage 0 of hy-

perasymptotics, i.e. optimal trunction of the series

(superasymptotics), ε ≈ 3.6 × 10−7; after stage 1,

ε ≈ 1.3 × 10−11; after stage 2, ε ≈ 4.4 × 10−14; after

stage 3, ε ≈ 6.1 × 10−15. At each stage, the error is

of the same order as the first neglected term.

ries. With each ‘hyperseries’ truncated at its leastterm, this incorporated all subdominant exponen-tials and all associated Stokes phenomena; andthe accuracy obtained far exceeded superasymp-totics (Figure 5) but was nevertheless limited.

It was clear from the start that in manycases unlimited accuracy could, in principle, beachieved with hyperasymptotics, by truncatingthe hyperseries not at the smallest term but be-yond it (although this introduces numerical sta-bility issues associated with the cancellation oflarger terms). This version of the hypersymptoticprogramme was carried out by Adri Olde Daal-huis, who reworked the whole theory, introducingmathematical rigor and effective algorithms forcomputing Dingle’s terminant integrals and theirmultidimensional generalizations, and applied thetheory to differential equations with arbitrary fi-nite numbers of transition points.

There has been an explosion of further devel-opments. Ecalle’s rigorous formal theory of resur-gence has been developed in several ways, basedon the Borel (effectively inverse-Laplace) trans-form. This converts the factorially-divergent se-ries into a convergent one, with radius of con-

vergence determined by singularities on a Rie-mann sheet. These singularities are responsiblefor the divergence of the original series, and forintegrals discussed above, correspond to the ad-jacent saddle points. In the Borel plane, com-plex and microlocal analysis allows the resurgencelinkages between asymptotic contributions to beuncovered and exact remainder terms to be es-tablished. Notable results include exponentiallyaccurate representations of quantum eigenvalues(R. Balian & C. Bloch, A. Voros, F. Pham, E. De-labaere); this inspired the work of the current au-thors on quantum eigenvalue counting functions,linking the divergence of the series expansion ofsmoothed spectral functions to oscillatory correc-tions involving the classical periodic orbits.

T. Kawai and Y. Takei in Kyoto have ex-tended ‘formally exact’, exponentially accurate,WKB analysis to several areas, most notably toPainleve equations. They have also developed atheory of ‘virtual turning points’ and ‘new Stokescurves’. In the familiar WKB situation, with onlytwo wave-like asymptotic contributions, Stokeslines emerge from classical turning points, andnever cross. With three or more asymptotic con-tributions, Stokes lines can cross in the complexplane at points where the WKB solutions are notsingular. Local analysis shows that an extra, ac-tive, ‘new Stokes line’ sprouts from one side onlyof this regular point; this can be shown to emergefrom a distant virtual turning point, where, un-expectedly, the WKB solutions are not singular.This discovery has been explained by C.J. Howls,P. Langman and A.B. Olde Daalhuis in terms ofthe Riemann sheet structure of the Borel planeand linked hyperasymptotic expansions, and in-dependently by S.J. Chapman and D.B. Mortimerin terms of matched asymptotics.

Groups led by S.J. Chapman and J.R. Kinghave developed and applied the work ofM. Kruskal and H. Segur to a variety of non-linear and PDE problems. This involves a lo-cal matched-asymptotic analysis near the distantBorel singularities that generate the factorially-divergent terms in the expansion, to identify theform of late terms, thereby allowing for an op-timal truncation and exponentially accurate ap-proach. Applications include: selection prob-lems in viscous fluids, gravity-capilliary solitary

7

waves, oscillating shock solutions in Kuramoto-Shivashinsky equations, elastic buckling, nonlin-ear instabilities in pattern formation, ship wavemodelling and the seeking of reflectionless hullprofiles. Using a similar approach, O. Costin andS. Tanveer have identified and quantified the ef-fect of ‘daughter singularities’ not present in ini-tial data of PDE problems, but which are gener-ated at infinitesimally short times. In so doing,they have also found a formally exact Borel rep-resentation for small-time solutions of 3D Navier-Stokes equations, offering a promising tool to ex-plore the global existence problem.

Other applications include quantum transi-tions, quantum spectra, the Riemann zeta func-tion high on the critical line, and even the philos-ophy of representing physical theories describingphenomena at different scales by singular rela-tions.

Further Reading

1. Batterman, R. W., 2002, The Devil in the De-tails: Asymptotic Reasoning in Explanation, Re-duction and Emergence (University Press, Ox-ford).

2. Berry, M. V., 1989, Uniform asymptotic smooth-ing of Stokes’s discontinuities Proc. Roy. Soc.Lond. A422, 7-21.

3. Berry, M. V. & Howls, C. J., 1991, Hyperasymp-totics for integrals with saddles Proc. Roy. Soc.Lond. A434, 657-675.

4. Delabaere, E. & Pham, F., 1999, Resurgentmethods in semi-classical mechanics Ann. Int.H. Poincare 71, 1-94.

5. Digital Library of Mathematical Functions,2010, NIST Handbook of MathematicalFunctions (University Press, Cambridge)http://dlmf.nist.gov, chapters 9 and 36.

6. Dingle, R. B., 1973, Asymptotic Expansions:their Derivation and Interpretation (AcademicPress, New York and London).

7. Howls, C. J., 1997, Hyperasymptotics for mul-tidimensional integrals, exact remainder termsand the global connection problem Proc. R. Soc.Lond. A453, 2271–2294.

8. Kruskal, M. D. & Segur, H., 1991, Asymptoticsbeyond all orders in a model of crystal growthStud. App. Math. 85, 129-181.

9. Olde Daalhuis, A. B., 1998, Hyperasymptotic so-lutions of higher order linear differential equa-tions with a singularity of rank one Proc. R.Soc. Lond. A454, 1-29.

10. Stokes, G. G., 1864, On the discontinuity of arbi-trary constants which appear in divergent devel-opments Trans. Camb. Phil. Soc. 10, 106-128.