Y=1

Boundary Layer

Inviscid Region

Flow

} Channel Midline Y=0

Figure 6. Schematic of the Berman-Terrill-Robinson problem. Fluid in the channel

¬‚ows to the right, driven partly by ¬‚uid pumped in through the porous wall. Only

half of the channel is shown because the ¬‚ow is symmetric with respect to the midline

of channel (dashed)

There is no ambiguity: far to the right, the WKB approximation must

approximate a transmitted, rightgoing wave and nothing else. This, in

an analysis too widely published to be repeated here, allows the analyti-

cal determination of β through standard WKB or matched asymptotics

expansions.

In contrast, standard WKB is quite impotent for determining the

di¬erence between the amplitude of the re¬‚ected wave and one because

the large re¬‚ected wave swamps the exponentially small correction.

However, ± is easily found indirectly by combining the known values

of the incoming and transmitted waves with conservation of energy.

Similarly, WKB gives a good approximation to the bound states and

eigenvalues of a potential well: where the wavefunction is exponentially

small (for large | x |), there is no competition from terms that are

larger.

A nonlinear example is the “Berman-Terrill-Robinson” or “BTR”

problem, which is interesting in both ¬‚uid mechanics and plasma physics

[135, 154, 108, 193, 186, 109]. In its mechanical engineering application,

the goal is to calculate the steady ¬‚ow in a pipe or channel with porous

walls through which ¬‚uid is sucked or pumped at a constant uniform

velocity V . Berman [23] showed that for both the pipe and channel,

the problem could be reduced to a nondimensional, ordinary di¬eren-

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31

Exponential Asymptotics

tial equation which in the channel case is

fY Y Y + fY ’ f fY Y = ±2

2

(48)

where ± is the eigenparameter which must be computed along with

f (Y ). The boundary conditions are

f (1) = 1, fY (1) = 0, f (0) = 0, fY Y (0) = 0 (49)

The small parameter is = 1/R where R is the usual hydrodynamics

“Reynolds number” (very large in most applications). Symmetry with

respect to the midline of the channel (at Y = 0) is assumed.

By matching asymptotic expansions, boundary layer to inviscid inte-

rior

(Fig. 6), one can easily compute a solution in powers of . Unfor-

tunately, the numerical work of Terrill and Thomas [292] showed that

there are actually two solutions for the circular pipe for all Reynolds

numbers for which solutions exist. Terrill correctly deduced that the

two modes di¬ered by terms exponentially small in the Reynolds num-

ber (or equivalently, in 1/ ) and analytically derived them in 1973 [291],

quite independently of all other work on hyperasymptotics.

The early numerical work on the porous channel was even more

confusing [265], ¬nding one or two solutions where there are actual-

ly three. Robinson resolved these uncertainties in a 1976 article that

combined careful numerical work with the analytical calculation of the

exponentially small terms which are the sole di¬erence between the two

physically interesting solutions.

The reason that the exponential terms could be calculated with-

out radical new technology is that the solution in the inviscid region

(“outer” solution) is linear in Y plus terms exponentially small in :

f (Y ) ∼ ±( )Y + γ( ) ’3 + Y 3 + ... (50)

±

1/4

1 2 1 1 5 253

γ( ) = ± exp ’ exp ’ 1’ ’ 2

+ O( 3 )

π7

6 4 4 4 32

(51)

(Note that because of the ± sign, there are two solutions for γ, re¬‚ecting

the exponentially small splitting of a single solution (in a pure power

series expansion) into the dual modes found numerically.) It follows

that by making the almost trivial change-of-variable

g ≡ f ’ ±Y (52)

we can recast the problem so that the “outer“ approximation is propor-

tional to exp(’1/(4 )). Systematic matching of the “inner” (boundary

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32 John P. Boyd

layer) and “outer” ¬‚ows gives the exponentially small corrections in

the boundary layer, too, even though there are non-exponential terms

in this region.

Other ¬‚uid mechanics cases are discussed in Notes 10 and 11 of the

1975 edition of Van Dyke™s book [298]. Bulakh[85] as early as 1964

included exponentially small terms in the boundary-layer solution to

converging ¬‚ow between plane walls and showed that such terms will

also arise at higher order in ¬‚ows with stagnation points. Adamson

and Richey[2] found that for transonic ¬‚ow with shock waves through

a nozzle, exponentially small terms are as essential as for the BTR

problem.

Happily, there is a widely-applicable strategy for isolating exponen-

tial smallness which is the theme of the next section. The key idea is

that the optimal truncation of the power series is always available to

rewrite t he problem in terms of a new unknown which is the di¬erence

between the original u(x; ) and the optimally-truncated series. Because

this di¬erence δ(x; ) is exponentially small in 1/ , we can determine it

without fear of being swamped by larger terms.

10. Darboux™s Principle and Resurgence

“Evidently, the determination of the remainder [beyond the superasymp-

totic approximation] entails the evaluation of several transcendental

functions. In other words, the calculation of the correction can be

more formidable than that of the original asymptotic expansion.

One is reminded of the dictum, sometimes asserted in physics, that

getting an extra decimal place demands 100 times the e¬ort expend-

ed on the previous one. Fortunately, the multiplying factor is not

so huge in our case but it is perforce appreciable.”

” D. S. Jones (1990) [155] [pg. 261]

Jones™ mildly pessimistic remarks are still true: hyperasymptotics is

more work than superasymptotics and one does have to evaluate addi-

tional transcendentals. However, Dingle showed in a series of articles in

the late ¬fties and early sixties, collected in his 1973 book, that there is

a suprising universality to hyperasymptotics: a quartet of generic tran-

scendentals su¬ces to cover almost all cases. The key to his thinking,

re¬ned and developed by Berry and Howls, Olver and many others, is

the following.

De¬nition 5 (Darboux™s Principle). One may derive an asymp-

totic expansion in degree j for the coe¬cients aj of a series solely from

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33

Exponential Asymptotics

knowledge of the singularities of the function f (z) that the series rep-

resents. This principle applies to power series [110, 111, 123, 82, 83],

Fourier, Legendre and Chebyshev series [55], and divergent power series

[118]

“Singularity” is a collective terms for poles, branch points and other

points where a complex functionf (z) ceases to be an analytic function

of z. If f (z) is singular, on the same Riemann sheet as the origin, at the

set of points {zj }, then the radius of convergence of the power series

for f (z) is ρ = min | zj |, as proven in most introductory calculus

courses. Darboux showed that if the convergence-limiting singularity

was such that f (z) = (z ’ zc )r g(z) where g(z) is nonsingular at the

convergence-limiting singularity, then the power series coe¬cients are

asymptotically (if j = integer)

aj ∼ j ’1’r zc {1 + O(1/j)}

’j

(53)

Asymptotics-from-singularities can be extended to logarithms and oth-

er singularities, too. As reviewed in [55], one can derive similar asymp-

totic approximations to the coe¬cients of Fourier, Chebyshev, Legendre

and other orthogonal expansions from knowledge of the singularities of

f (z).

Dingle [116, 117] realized in the late 50™s that Darboux™s Principle

applies to divergent series, too. If one makes an asymptotic expansion

by performing a power series expansion inside an integral and then

integrating term-by-term, the coe¬cients of the divergent expansion

will be simply those of the power series in the integration variable

multiplied by the e¬ect “ usually a factorial “ of the term-by-term

integration. For example, consider the class of functions

∞

f( ) ≡ exp(’t) ¦( t)dt (54)

0

where ¦(z) has the power series

∞

bj z j

¦(z) = (55)

j=0

then

∞

f( ) ∼ aj j ; aj = j! bj (56)

j=0

Because the coe¬cients of the divergent series {aj } are merely those of

the power series of ¦, multiplied by j!, it follows that the asymptotic

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34 John P. Boyd

behavior of the coe¬cients of the divergent series must be controlled by

the singularities of ¦(z) as surely as those of the power series of ¦ itself.

In particular, the singularity of the integrand which is closest to t = 0

must determine the leading order of the coe¬cients of the divergent

expansion. This implies that all f ( ) that have a function ¦(z) with

a convergence-limiting singularity of a given type (pole, square root,

etc.) and a given strength (the constant multiplying the singularity)

at a given point zc will have coe¬cients that asymptote to a common

form, even if the functions in this class are wildly di¬erent otherwise.

EXAMPLE: The “double Stieltjes” function

SD( ) ≡ S( ) + S( /2) (57)

where S( ) is the Stieltjes function described earlier. The asymptotic

series is

∞

1

SD( ) ∼ aj j ; aj = (’1)j j! 1 + j (58)

2

j=0

The integrand of S( ) is singular at t = ’1/ while that of S( /2) is

singular twice as far away at t = ’2/ . In the braces in Eq. (58), the

¬rst and nearer singularity contributes the one while the rapidly decay-

ing factor 1/2j comes from the more distant pole of the integrand, that

of S( /2). The crucial point is that in the limit j ’ ∞, the coe¬cients

of the divergent series for the double Stieltjes function asymptote to

those of the ordinary Stieltjes function.

As explained above, the optimal truncation of the power series

for the Stieltjes function is to stop at N = [1/ ], that is, at the integer

closest to the reciprocal of ; the error in the resulting “superasymptot-

ic” approximation is proportional to exp(’1/ ). The dominance of the

asymptotic coe¬cients of the double Stieltjes function by the pole at

t = ’1/ implies that all these conclusions should apply to the optimal

truncation of the divergent expansion for SD( ),too:

Nopt

SD( ) ∼ j

aj +O π/(2 ) exp(’1/ ) ; Nopt ( ) = [1/ ]

j=0

(59)

where the factor in front of the exponential is justi¬ed in [19]. More

important, if we add the error integral for Stieljes function to the

Nopt ( )-term truncation of the series for the double Stieltjes function,

we should obtain an improved approximation. Since the ¬rst neglected

term in the series for SD( ) di¬ers from that included in the Stieltjes

error integral by a relative error of O( 2N ), the best we can hope for is

to improve upon the superasymptotic approximation by a factor of 2N ,

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35

Exponential Asymptotics

0

10

’2 Supe

10 rasy

mpto

tic

N

’4 po

10 we

rs