2

’6 N erm

10 po ina

we

nt

rs

’8 +

10 Te

rm

ina

’10 nt

10

0 2 4 6 8 10 12

1/µ

Figure 7. Double Stieltjes function: errors in three approximations. x™s: Errors in

optimally-truncated asymptotic series (the “superasymptotic” approximation. Plus-

es: Superasymptotic approximant plus the “terminant”. Circles: Approximation

de¬ned by Eq. (55). Solid curves: Predicted errors, which are respectively the fol-

lowing ” (top) q exp(’1/ ), (middle) q exp(’1.693/ ), (bottom) q exp(’2/ ) where

q( ) ≡ (π/(2 ))1/2 .

which, because Nopt ≈ 1/ , can be rewritten as exp(’ log(2)/ ). Thus,

N

SD( ) ∼ j

aj + EN ( ) + O (exp(’{1 + log(2)}/ )) ; N ( ) = [1/ ]

j=0

(60)

where EN ( ) is the error integral for the Stieltjes function de¬ned by

Eq.(8). Fig. 7 shows that the error estimate in Eq. (60) is accurate.

If the location of the second-worst singularity is known ” that is,

the pole or branch point of the integrand which is closer to t = 0 than

all others except the one which asymptotically dominates “ one can do

better. Since the second pole of SD( ) is at twice the distance of the

¬rst, if we add the next N contributions of the second singularity only

only to the approximation of Eq. (60), the result should be as accurate

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36 John P. Boyd

as the optimal truncation of a series derived from the second singularity

(i. e., S( /2) for this example), that is, have an error proportional to

exp(’2/ ):

2N

N

j! 2

SD( ) ∼ + O exp ’

j

(’1)j j

aj + EN ( ) + (61)

2j

j=0 j=N +1

Fig. 7 con¬rms this. (Howls[147] and Olde Daalhuis[241] have devel-

oped improved hyperasymptotic schemes with smaller errors, but for

expository purposes, we have described the simplest approach.)

A key ingredient in Dingle™s strategy is Borel summation. Under

certain conditions [318], a divergent series can be summed by the inte-

gral of exp(’t) multipied by a function ¦( t) which is de¬ned to be

that function whose power series has the coe¬cients of the divergent

series divided by j!. That is to say, the integral in (54) is the Borel

sum of the power series for the function f ( ) on the left in the same

equation. (We are again reminded of the interplay between di¬erent

strategies in hyperasymptotics; a series acceleration method, which is

a hyperasymptotic method in its own right when combined with Pad´ e

approximation of ¦( t) [“Pad´-Borel” method [315, 316]], is also a key

e

justi¬cation for a di¬erent and sometimes more powerful hyperasymp-

totic scheme.) Dingle™s twist is that he applies Borel summation only to

the late terms in the asymptotic series. The ¬rst few terms in the sum

for SD( ) are very di¬erent from those of the Stieltjes function; the only

way to obtain the right answer is to sum these leading terms directly

without tricks. Dingle™s key observation is that the late terms, meaning

those neglected in the optimal truncation, are essentially the same as

those for the ordinary Stieltjes function. Thus, the error integral EN ( )

for one function, S( ), provides a hyperasymptotic approximation to

an entire class of functions, namely all those of the form of Eq. (54) for

which the convergence-limiting singularity of ¦(z) is a simple pole at

z = ’1.

It might seem as if we would have to repeat the analysis for each

di¬erent species of singularity ” one family of error integrals when the

singularity is a simple pole, another when the dominant singularity of

¦ is a logarithm and so on. In reality, Dingle shows that for a very wide

range of asymptotic expansions, both from integral representations, the

WKB method, and so on, the coe¬cients are asymptotically of the form

“(j + 1 ’ β)

aj ∼ q(’1)j (62)

ρj+1’β

for some constants q, ρ and β. The error integral for the Stieltjes func-

tion is almost the theory of everything.

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37

Exponential Asymptotics

In the next three sections, we describe how Dingle™s theory has been

extended to the method of steepest descent and the mystery of Stokes

phenomenon. A couple of historical, semantic, and notational grace

notes are needed ¬rst, however.

The ¬rst is that the work of Dingle and others is couched not in

terms of the error integrals EN ( ) but rather in terms of the following:

De¬nition 6 (Terminant). A function TN ( ) is a “terminant” if

it is used to weight the N -th term in an asymptotic series so as to

approximate the exact sum .

The reason for working with terminants instead of errors is mostly

historical. Stieltjes [286] showed that for an alternating series, one could

considerably improve accuracy for both convergent and divergent series

merely by multiplying the last retained term by a weight factor of 1/2.

Airey developed an early (1937) hyperasymptotic method, restricted to

alternating series for which the general term is known, which comput-

ed an improved, N -dependent replacement for Stieltjes™ 1/2 [3]. Later

studies have generally followed this convention. However, terminants

are sometimes more convenient than error integrals as in the smooth-

ing of Stokes phenomenon.

The second comment is that Dingle found it helpful to de¬ne four

canonical (approximate) terminants instead of one. One reason is that

the Stieltjes error integral, and the equivalent terminant, have poles on

the negative real axis away from the integration interval, which is the

positive real axis. Stokes phenomenon happens when the poles coincide

with integration interval, which makes it convenient to de¬ne a second

terminant. Dingle™s fundamental pair are

∞ exp(’t) tm

1

Λm (1/ ) ≡ dt (63)

“(m + 1) 1+ t

0

∞ exp(’t) tm

1

Λm (1/ ) ≡ P dt (64)

1’ t

“(m + 1) 0

where P denotes the Cauchy Principal Value of the integral. These two

¬ssion into two more because many expansions proceed in powers of 2

rather than itself, which makes it convenient to de¬ne terminants for

even powers of , his Πm and Πm .

Furthermore, newer classes of problems have required additional ter-

minants, as illustrated in Delabaere and Pham[113]. When the hyper-

asymptotic process is iterated so as to add additional terms, with di¬er-

ent scalings, one needs generalizations of the Dingle terminants called

“hyperterminants”. Olde Daalhuis[240, 242] has given algorithms for

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38 John P. Boyd

the numerical computation of terminants and hyperterminants. The

need for these generalizations, however, should not obscure the funda-

mental unity of the idea of adding error integrals or terminants that

match the dominant singularity to convert a superasymptotic approx-

imation into a hyperasymptotic approximation.

There is a close parallel between Dingle™s universal terminants for

asymptotic series and the universal error envelopes for Chebyshev and

Fourier spectral methods which were derived by Boyd [55, 59]. For

example, Boyd found that the error envelope was always a linear com-

bination of the same two meromorphic functions (the “Lorentzian” and

“serpentine” functions, de¬ned in [59]), regardless of whether the func-

tion being interpolated was entire, meromorphic, or had logarithmic

singularities. Even when f (x) is nonanalytic but in¬nitely di¬erentiable

at a point on the expansion interval, and thus has only a divergent

power series about that point, the error envelope is the sum of these

two functions. The reason for the similarity is that Darboux™s Principle

applies to Fourier and Chebyshev series, too. Asymptotically, functions

that are very dissimilar in their ¬rst few terms resemble each other more

and more closely in the late terms. One or two terminants can encap-

sulate the error for very di¬erent classes of functions, even ones whose

late coe¬cients are decaying, because of the magic of Taylor expansions

with respect to degree.

11. Steepest Descents

“The resultant series is asymptotic, rather than convergent, because

the range of integration extends beyond the circle of convergence of

[the power series of the metric factor], the radius of this circle being

¬xed by the zero of dφ/dt in the complex w-plane lying closest to

the origin.”

” R. B. Dingle [118], pg. 111, with translation of nota-

tion into the symbols used in the section below.

The method of steepest descent is commonly applied to evaluate the

integral

I(z) ≡ exp(zφ(t))dt (65)

in the limit | z |’ ∞. As described in standard texts [19], the “saddle

points” or “stationary points” {ts } play a crucial role where these are

de¬ned as the roots of the ¬rst derivative of the “phase function” φ(t):

dφ

(ts ) = 0 (66)

dt

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39

Exponential Asymptotics

The path of integration is deformed so as to pass through one or more

saddle points. The next step is to identify the dominant saddle point,

which is the one on the deformed contour of integration for which

(φ(ts )) is largest. Restricting ts to the dominant saddle point, one

then makes the exact change-of-variable

w≡ φ(ts ) ’ φ(t) (67)

so that the integral becomes

dt

exp(’zw2 )

I(z) = (w) dw (68)

dw

The ¬nal steps are (i) extend the integration interval to the entire real

w-axis and (ii) expand the “metric factor” dt/dw in powers of w and

integrate term-by-term to obtain an exponential factor multiplied by an

inverse power series in the large parameter z. (By setting = 1/z, this

series is similar “ and similarly divergent “ to the power series explored

earlier.) We omit details and generalizations because the mechanics are

so widely described in the literature[19, 319].

Unfortunately the standard texts hide the fact that the asymptotic

expansion is based on the same mathematical atrocity as the divergent

series for the Stieltjes function: employing a power series in the inte-

gration variable with a ¬nite radius of convergence under integration

over an in¬nite interval. Hyperasymptotics is greatly simpli¬ed by the

following.

Theorem 1 (Singularities of the Steepest Descent Metric Function).

If an integral of the form of Eq.(65) is transformed by the mapping

Eq. (67) into the integral over w, Eq. (68), then the metric factor dt/dw

has branch points of the form

dt g(w)

=√ + h(w) (69)

w ’ ws

dw

where g(w) and h(w) are analytic at w = ws . All such points ws are

the images of the saddle points ts under the mapping w(t); conversely,

the metric factor is singular at all points ws which are images of saddle

points except for w = 0. The metric factor may also be singular with

singularities of more complicated type at points w which are images of

points where the “phase factor” φ(t) is singular.

PROOF: The ¬rst step is to di¬erentiate the de¬nition of the map-

ping Eq.(67) to obtain

dw 1 dφ dt w

=’ ⇐’ = ’2

; (70)

2 φ(ts ) ’ φ(t) dt

dt dw dφ/dt

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40 John P. Boyd

This shows that the metric factor can be singular only at the w-images

of those points t in the original integration variable where (i) φ(t) is

singular or (ii) saddle points where by the very de¬nition of a saddle

point, dφ/dt = 0 and the denominator of the right-hand side of Eq.(70)

is zero. This is really just a restatement of the implicit function of