then the inverse function t(w) exists and is analytic at that point and
its derivative dt/dw = 1/(dw/dt). The point w = 0 is exceptional
because the numerator of the righthand side (w) cancels the zero in
the denominator.
To obtain an expression for dt/dw in the neighborhood of a saddle
point, we expand w(t) about the saddle point t = ts . The constant ws
can be moved to the left side of the equation and the linear term is
zero because dw/dt is zero at the saddle point. Taking the square root
gives
√ 1 d2 w
w ’ ws = (t ’ ts ) (ts ) {1 + O(t ’ ts )} (71)
2 dt2
√
It follows that dt/dw is proportional to 1/ w ’ ws near the saddle
point, which demonstrates the theorem.
Denote the imageofasaddlepoint of smallest absolute value by
wmin . The coe¬cients bj of the power series of the integrand will
then asymptote, for su¬ciently high degree j, to those of a constant
√
times 1/ w ’ wmin ; the contributions of the singularities that are more
remote in the complex wplane will decrease exponentially fast with
j compared to the contribution of the square root branch point at
w = wmin . Applying the binomial theorem to compute the power series
coe¬cients of the square root singularity and then integrating term
byterm shows that the coe¬cients aj of the asymptotic series for the
integral itself will asymptote for large j to
“(j + 1/2)
aj ∼ q (72)
 wmin j+1/2
where the constant q is proportional to g(wmin ) in the theorem. Dingle
[118], pg. 457, gives the basic terminant (with some changes in notation)
TN ∼ q0 ΛN ’1 (’F ) + q2 ΛN ’2 (’F ) + q4 ΛN ’3 (’F ) + . . . (73)
where the q2j are functions of Dingle™s “chief singulant” F , which in
our notation is
F ≡ zwmin
2
(74)
and q2j ∼ O(F j ) This situation is more complicated than for the dou
ble Stieltjes function in that we have a series of terminants, rather
ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.40
,
y
yy
zy,
,z
,
z
y z
41
Exponential Asymptotics
zz
,
y
y,
zy
,,
,y
yy
z

,,
z,,
AntiStokes line Stokes line
,,,
yyy
zyzz
z
y

z


Figure 8. Stokes Lines (Monotonic Growth/Decay) and anti“Stokes Lines (Pure
Oscillation) for the Airy Functions Ai and Bi. The shaded regions show the transition
zone for the Stokes™ multiplier of Ai, that is, the regions where it varies from 1 to 0
as an error function. The positive real axis is a Stokes Line for Bi but not Ai. The
shaded regions narrow for large z because for the Airy function, the width of the
transition zone, expressed in terms of the angle θ ≡ arg(z), decays as z’3/4 .
than a single terminant. (Each term of the expansion of dt/dw in half
integral powers of w ’ wmin will generate its own contribution to the
terminant series.) The underlying ideas remain simple even though the
algebraic complexity rapidly leaves one muttering: “Thank heavens for
Maple! [and similar symbolic manipulation languages like Mathemati
ca, Reduce and so on].”
12. Stokes Phenomenon
“ about the present title [Divergent Series], now colourless, there
hung an aroma of paradox and audacity.”
” Sir John E. Littlewood, (18851977)[139]
Stokes phenomenon has contributed much to the “aroma of paradox
and audacity” of asymptotic series. It is easiest to explain by example.
The Airy function Ai(z) asymptotes for large positive z to the prod
uct of a decaying exponential with a series in inverse powers of z 3/2 . For
ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.41
42 John P. Boyd
θ= 2 π/3 θ= π/3
5 5
„‘(t)
„‘(t)
0 0
5 5
5 0 5 5 0 5
θ= π θ=0
5 5
„‘(t)
„‘(t)
0 0
5 5
5 0 5 5 0 5
„(t) „(t)
Figure 9. Steepest descent paths of integration in the complex plane of the original
integration variable t for four di¬erent values of z. The two saddle points are marked
by black discs. The contours of log( exp(z 3/2 i t + t3 /3 ) ) from are also shown
For θ = π [negative real zaxis, lower left panel], the integration contour comes from
large t in the upper right quadrant, returns to in¬nity along the negative imaginary
taxis, and then returns to pass through the right saddle point and depart to in¬nity
via the lower right tquadrant.
negative real z, the Airy function is real and oscillatory; approximated
by the product of a cosine with a inverse power series plus a sine with
a di¬erent inverse power series. However, the multiplier of the leading
inverse power, the cosine, is the sum of two exponentials. If we track
the asymptotic approximation for ¬xed  z  as θ = arg(z) varies from
0 to π, one exponential must somehow metamorphosize into two.
The classical analysis hinges on two species of curves in the complex
zplane: “Stokes lines”, where the exponentials grow or decay without
oscillations, and “antiStokes” lines where the exponentials oscillate
without change in amplitude.1 (Fig. 8.) Stokes™ own interpretation is
1
We employ the convention of Heading, Dingle, Olver, and Berry, but other
authors such as Bender and Orszag reverse the meaning of “Stokes” and “anti
Stokes”.
ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.42
43
Exponential Asymptotics
5
0
5
10
2
5
0 0
2 5
„‘(t) „(t)
Figure 10. A surface plot of log( exp(z 3/2 φ)) for the Airy integral for arg(z) =
2π/3, that is, on the Stokes line. The steepest descent path is marked by the heavy
solid line; the disks denote the two saddle points at ts = ±i. The surface has been
truncated at the vertical axis limits for graphical clarity.
that the coe¬cient of the “recessive” (decaying) exponential jumps dis
continuously on the Stokes line (for Ai(z), at arg(z) = ±2π/3), that
is, where this exponential is smallest relative to the “dominant” expo
nential that grows as  z  increases along the Stokes line. As the nega
tive real axis (an antiStokes line) is approached, the two exponentials
become more and more similar until ¬nally both are purely oscillatory
with coe¬cients of equal magnitude on the antiStokes line itself.
The annoying and unsatisfactory part of this discontinuous jump is
that the Airy function itself is an entire function, completely free of
all jumps, in¬nities and pathologies of all kinds except at  z = ∞.
Sir Michael Berry has recently smoothed this “Victorian discontinu
ity”, to quote from one of his papers, by combining Dingle™s ideas with
the standard and longknown asymptotic approximation to an inte
gral when the saddle point and a pole nearly coincide. To understand
Berry™s jumpfree hyperasymptotics, we need some preliminaries.
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44 John P. Boyd
First, let us represent the solution by an integral which can be
approximated by the method of steepest descent for large z. (Berry™s
smoothing is equally applicable to WKB approximations to di¬erential
equations and a wide variety of other asymptotics, but steepest descent
is the most convenient for explaining the concepts.)
The integral representation for the Airy function is
z 1/2
Ai(z) ≡ exp z 3/2 i (t + t3 /3) dt (75)
2π C
where C is a contour that originates at in¬nity at an angle arg(t) =
(5/6)π ’ (1/2) arg(z) and returns to in¬nity at arg(t) = (1/6)π ’
(1/2) arg(z).
Fig. 9 shows the steepest descent paths of integration for the Airy
integral representation. As explained in the preceding section, the eas
iest way to generate the coe¬cients of the asymptotic series is to begin
with a changeofcoordinate to a new integration variable w. To illus
trate the key topological ideas, however, it is perhaps more illuminating
to illustrate the steepest descent path in the t ’ plane as we have done
in the ¬gure. In either plane, the path of integration is deformed so as
to pass through a saddle point, and then curve so that at each point t
on the path, (zφ(ts )) = (zφ(ts )). This condition that the phase of
the integrand matches that of the saddle point ensures that the mag
nitude of the integrand decreases as steeply as possible from its local
maximum, i. e., that the curve is really is a path of “steepest descent”.
As a student, I was much puzzled because my texts and teachers
expended a lot of energy on determining the exact shape of the steepest
descent contour even though it does not appear explicitly in the answer,
even at higher order! The steepest descent path is actually important