lie on the path, but the shape of the contour is otherwise irrelevant.

For the Airy function, for example, there are two saddle points for

all z but only one is on the contour for large positive z. As the argument

of z varies, however, the steepest descent paths in the t (or w) planes

must vary also. For some z, the steepest descent path through one

saddle point must collide with the other; this happens precisely on the

Stokes lines.

As shown in Fig. 9, the Stokes lines are a change in the topology of

steepest descent paths: a single saddle point on the contour on one side

of the Stokes line, two saddle points on the other side of the Stokes

line and on the Stokes line itself. Thus, Berry™s title for one of his arti-

cles, “Smoothing a Victorian discontinuity”, is a bit misleading since

the discontinuity is not removed in a topological sense. The jump is,

however, smoothed numerically.

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45

Exponential Asymptotics

Parenthetically, note that at the Stokes line itself (arg(z) = 2π/3

for Ai(z)), the steepest descent path descends from one saddle point

directly to another saddle point, then makes a right angle turn and

then continues to descend monotonically from the second saddle point

(Fig. 10). For arg(z) > 2π/3, the steepest descent contour from one

saddle point does not runs o¬ to in¬nity parallel to the negative imagi-

nary axis. To be continuous and still terminate at ∞ exp(iπ/6), howev-

er, the contour must return and pass through the second saddle point.

At arg(z) = π, the contributions of both saddle points are equal.

The properties of Stokes lines may be summarized as follows:

1. There are TWO saddle points on the steepest descent integration

path in the t-plane.

{z(φ(t+ ) ’ φ(t’ )} = 0 where t+ and t’ are the two saddle points

2.

on the steepest descent contour and where φ(t) is the steepest

descent phase function de¬ned by Eq. 65.

3. The terminants for the series each have a simple pole on the real

w-axis, which is the integration interval after the usual steepest

descent change of variable, the poles being at the saddle point

which contributes the “recessive” saddle point.

4. The terms bj of the asymptotic inverse power series are, for su¬-

ciently large degree j, all of the same sign.

When there is a discontinuity in asymptotic form, the ¬rst three prop-

erties are each equivalent de¬nitions of a “Stokes line”.

The proofs of these assertions and also generalizations of Stokes phe-

nomenon to solutions of nonlinear di¬erential equations and so on are

´

given by the theory of “resurgence”. Ecalle[123] invented “resurgence”

[123] and the formalism of the “alien calculus” and “multisummabil-

ity”. This has been extended by a group that includes Voros, Pham,

Sternin, Shatalov, Delabaere, and others too numerous to list. The

monograph by Sternin and Shatalov[285] and the collection of articles

edited by Braaksma[83] are good summaries. (Berry, who was visiting

Pham when he developed his smoothing scheme, was strongly in¬‚u-

´

enced by Ecalle™s three-volume book and the follow-up work of the

“French school”.) The alien calculus and multisummability theory are

very general but accordingly also very abstract. Berry and Howls, Olde

Daalhuis and Olver, Costin, Kruskal, Hu and others have developed

simpli¬ed variants of resurgence and applied them to concrete prob-

lems in special functions and physics.

As shown by the sheer length of the Table IV, which is a selected

bibliography of works on resurgence and Stokes phenomenon, it is quite

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46 John P. Boyd

´

unfeasible to summarize this powerful theory here. (Prof. Ecalle™s pio-

neering treatise is in three volumes!) Still, one can give a little of the

¬‚avor of resurgence.

One key concept is what one might call “saddle point democracy”.

Instead of focusing in quickly on one or two dominant saddle points

(on the steepest descent path), resurgence treats all saddle points on

an equal footing. One may de¬ne an integral passing through an arbi-

trary saddle point; the coe¬cients of the steepest descent expansion

about that point encodes the expansions about all the other saddle

points. Furthermore, the late terms in the asymptotic expansion about

a dominant saddle point can be expressed in terms of the early terms of

a subdominant series, and vice-versa. The reason is that the late terms

in the expansion about the dominant saddle point are controlled, via

Darboux™s Principle, by the singularities created by the other saddle

points.

13. Smoothing Stokes Phenomenon: Asymptotics of the

Terminant

“Having these new techniques [hyperasymptotics], I would like to

hear from anybody who needs the Airy function to twenty decimals,

but am not expecting an early call.”

” Berry (1991) [30] [pg. 2.]

Berry™s amusing comment is a frank admission that the smooth-

ing of the discontinuity along a Stokes line is not a matter of great

arithmurgical signi¬cance. The term that changes dramatically in the

neighborhood of the Stokes line is exponentially small compared to the

sum of the asymptotic series. However, the smoothing does provide

deep insights into the interlocking systems of caverns ” interlocking

systems of expansions about di¬erent saddle points and branch points

” that lie beneath the surface of asymptotic approximations.

The numerical smoothing of the discontinuity along a Stokes lines

is based on the following ideas which will be explained below:

1. The exponentially small Stokes multiplier M can be isolated by

subtracting the optimal truncation of the standard asymptotic

series for f (z) from it so that the multiplier is no smaller than

the other terms left after the subtraction.

2. The subdominant saddle point, the one whose Stokes multiplier is

to change, lies directly on the steepest descent path leading down

from the dominant saddle point when z or is on the Stokes line.

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47

Exponential Asymptotics

3. When the asymptotic approximation for f (z) is optimally trun-

cated, the saddle point of the integral representation of Dingle™s

terminant will coincide with the subdominant saddle point and

therefore with the pole of the integrand

4. The method of steepest descent, applied to the integrand of the

terminant, replaces the integrand™s sharp peak at its saddle point

with a Gaussian function, thereby reducing the asymptotics of the

terminant to that of a Gaussian divided by a simple pole at the

origin.

5. If we allow the small parameter or the equivalent large parameter,

z = 1/ , to move a little way δ o¬ the Stokes line, the terminant

integral becomes the Fourier transform of a Gaussian divided by a

pole at (or very near) the maximum of the Gaussian

6. The Fourier transform of a Gaussian divided by a pole is that of

the integral of the Fourier transform of the Gaussian, which is the

error function erf.

To illustrate these ideas, de¬ne the “singulant” F via

F ≡ {z(φ(t+ ) ’ φ(t’ ))} (76)

where t+ and t’ are the two saddle points on the steepest descent

contour and where the aj are the coe¬cients of the inverse power series.

(The real part of the di¬erence between zφ(t) at the two points is zero

along a Stokes line, and this can be used to de¬ne a Stokes line.) The

singulant is proportional to some positive power of the large parameter

z so that the inverse power series in z can be expressed as inverse

powers of F .

The Stokes multiplier M may then be de¬ned by

±

Nopt (z)

aj F ’j+β’1 + iM σ’ (z) exp(’F )

f (z) ∼ exp(zφ(t+ )) σ+ (z)

j=0

(77)

where Nopt denotes the optimal truncation of the asymptotic series for

a given z, σ± (z) are slowly varying factors of z (usually proportional

to a power of z rather than an exponential), β depends on the class

of asymptotic approximation, and the coe¬cients have been scaled so

that a0 = 1 by absorbing factors into σ± if necessary. (For steepest

descent as discussed here, β = 1, but other values do occur when the

integral involves a contribution from an endpoint of integration interval

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48 John P. Boyd

or certain other classes of asymptotics[25].) This de¬nition is equivalent

to

±

exp(F )

Nopt (F )

aj F ’j+β’1

M ≡ ’i f (z) exp(’zφ(t+ )) ’ σ+ (z)

σ’ (z)

j=0

(78)

Replacing f (z) exp(’zφ(t+ )) by the in¬nite asymptotic series and sub-

tracting

±

σ+ (z)

∞

aj F ’j+β

M ∼ ’ i exp(F ) (79)

σ’ (z)

j=Nopt (F )+1

Note F is real and positive on the Stokes line.

The next step is to sum the series for the Stokes multiplier via Borel

summation. The follow-up is crucial: instead of employing the exact

power series coe¬cients aj in the Borel sum, we use the asymptotic

approximation to them as j ’ ∞. This is legitimate since only late

terms, i.e., those for j > Nopt (F ), appear in the sum. This approximates

the Stokes multiplier in terms of Dingle™s singular terminant ΛN (F ).

To illustrate this general strategy, we shall return to the speci¬c

example of the Airy function, which has the asymptotic approximation

1

Ai(z) ∼ z ’1/4 √ {E’ + i M E+ } (80)

2π

where

∞

exp(±(2/3)z 3/2 ) “(n + 5/6)“(n + 1/6)

E± ≡ (81)

“(n + 1) ( F )n

“(5/6)“(1/6) n=0

The Stokes™ multiplier M is zero when arg(z) = 0 and is unity when

arg(z) = π.

Dingle™s singulant is de¬ned by

4

F ≡ ’ z 3/2 (82)

3

which as always is the di¬erence between the arguments of the two

exponentials in the asymptotic approximation. Note the sign conven-

tion: F is negative when z and ζ are real and positive.

If the coe¬cients of the asymptotic series for E’ , which is the dom-

inant exponinant exponential near the Stokes line at arg(z) = (2/3)π,

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49

Exponential Asymptotics

are denoted by aj , then the argument given above implies that

±

∞

aj F ’j

M ∼ ’ i exp(F ) (83)

j=Nopt (F )+1

±

∞

1 ’j

∼ ’ i exp(F ) (j ’ 1)! F (84)

2π

j=Nopt (F )+1

∞

i 1

∼’ exp(F (1 ’ t)) tNopt dt (85)

1’t

2π 0

In the second line, we have replaced the aj by their asymptotic approx-

imation as j ’ ∞ [derived through the large degree asymptotics of

the gamma functions plus the identity “(1/6)“(5/6) = 2π]. The third

line was derived from the second by taking the Borel sum of the series,

which happens to be an integral with an integrand that can be written

down explicitly. We can check that the integral is correct by expanding

the integrand about t = 0 and then integrating term-by-term. The inte-