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+T
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):

(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

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