On the summation of divergent, truncated,
and underspeciﬁed power series
via asymptotic approximants
Nathaniel S. Barlow1, Christopher R. Stanton1, Nicole Hill2,
Steven J. Weinstein2, and Allyssa G. Cio3.
1School of Mathematical Sciences,2Department of Chemical Engineering,
3Department of Industrial Engineering
Rochester Institute of Technology, Rochester, NY 14623, USA
This article has been accepted for publication in the Quarterly Journal of
Mechanics and Applied Mathematics Published by Oxford University Press.
The ﬁnal published version can be found here: https://academic.oup.com/
A compact and accurate solution method is provided for problems
whose inﬁnite power series solution diverges and/or whose series coef-
ﬁcients are only known up to a ﬁnite order. The method only requires
that either the power series solution or some truncation of the power
series solution be available and that some asymptotic behavior of the
solution is known away from the series’ expansion point. Here, we
formalize the method of asymptotic approximants that has found re-
cent success in its application to thermodynamic virial series where
only a few to (at most) a dozen series coeﬃcients are typically known.
We demonstrate how asymptotic approximants may be constructed
using simple recurrence relations, obtained through the use of a few
known rules of series manipulation. The result is an approximant that
bridges two asymptotic regions of the unknown exact solution, while
maintaining accuracy in-between. A general algorithm is provided to
construct such approximants. To demonstrate the versatility of the
method, approximants are constructed for three nonlinear problems
relevant to mathematical physics: the Sakiadis boundary layer, the
Blasius boundary layer, and the Flierl-Petviashvili monopole. The
power series solution to each of these problems is underspeciﬁed since,
in the absence of numerical simulation, one lower-order coeﬃcient is
not known; consequently, higher-order coeﬃcients that depend recur-
sively on this coeﬃcient are also unknown. The constructed approx-
imants are capable of predicting this unknown coeﬃcient as well as
other important properties inherent to each problem. The approxi-
mants lead to new benchmark values for the Sakiadis boundary layer
and agree with recent numerical values for properties of the Blasius
boundary layer and Flierl-Petviashvili monopole.
Power series arise in virtually all applications of mathematical physics. The
utility of such series is evident in the construction of approximations (e.g.
ﬁnite diﬀerences) or as a rigorously determined solution to a problem. Lim-
itations generally inherent to power series solutions often inhibit their di-
rect use, and they are more often useful in the implementation of numerical
schemes. For instance, a Taylor series representation of an unknown function
may not converge, as it may have a ﬁnite radius of convergence arising from
singularities (often complex) in the function it represents . Even when
singularities are not a concern, higher-order terms of the series may be ex-
ceedingly diﬃcult to compute , which is especially problematic if the series
Several techniques have been put forward to analytically continue and
to accelerate convergence of divergent or slowly converging series; see, for
example  (ch. 8),  (chs. 14, 19, 20), , , , , , . One
of the more well-known techniques is the Pad´e approximant method ,
which has similarities with the method of asymptotic approximants described
herein. A Pad´e approximant (referred to as a Pad´e) is the quotient of two
polynomials, with Ntotal polynomial coeﬃcients (distributed between the
numerator and denominator), chosen such that the Taylor expansion of the
Pad´e reproduces the power series of interest up to Nterms. This involves
solving an algebraic system with Nunknowns (the Pad´e coeﬃcients). After
its construction, one may use the Pad´e in lieu of the original series. The Nth-
order Pad´e and Nth-order truncated original series will follow one another
near the expansion point. Ideally, the Pad´e will then continue on beyond
the radius of convergence (if one is present) and in general represent the
actual solution better than the truncated series. If chosen judiciously, a Pad´e
sequence (speciﬁed by a ﬁxed diﬀerence in order between the numerator and
denominator) may converge rapidly to the correct solution as Nis increased,
whereas the original series may converge slowly or diverge. In the case of
a divergent series, the Pad´e approximates the singularity that is presumed
to be responsible for divergence , and in doing so enables an accurate
summation of the truncated series.
One drawback of the Pad´e method is that it is not always clear before-
hand which sequence is best suited for a given problem . If one has a
power series alone, it is diﬃcult to choose a correct Pad´e form (see [13, 14, 15]
for example). However, if the power series arises from a physical problem, it
is likely that some additional conditions or context may be gleaned. These
conditions may be conveniently available, or may need to be independently
derived. For instance, if one has a power series representation of f(x) and
it is known that fasymptotically approaches a constant value as x→ ∞,
a “symmetric” Pad´e sequence of equal order in the numerator and denom-
inator will contain Pad´es that all preserve this asymptotic condition, and
this sequence will uniformly converge towards the correct solution as Nis
increased. Baker and Gammel  recognized this important result and went
further to state: If a series for f(x) is known and f∼Cxpas x→ ∞ (pbeing
an integer), the uniformly convergent Pad´e sequence is the one with a diﬀer-
ence of pbetween the numerator and denominator order, hence preserving
the x→ ∞ behavior as Nis increased.
Baker & Gammel’s statement may be extended to asymptotic behaviors
beyond integer power laws using approximants other than Pad´es [16, 17, 18].
An example of such an approximant is found in the review by Frost and
Harper , used to ﬁnd an approximate solution for the drag coeﬃcient
on a sphere in ﬂuid-ﬁlled tube; the approximant they used incorporates the
correct non-integer power-law asymptotic behavior. The present work may
be considered an extension of the perspective given in - namely the con-
struction of approximants that are asymptotically consistent with known
behavior in the vicinity of the domain boundaries while maintaining accu-
racy in-between. In this paper, we aim to formalize this approach by deﬁning
asymptotic approximants in general:
Deﬁnition 1.1 Given a power series representation of some function f(x):
and an asymptotic behavior
where Cis a constant, an asymptotic approximant is any function fA(x)that
may be expressed analytically in closed form and that satisﬁes the following
1. The N-term Taylor expansion of fAabout x0is identical to the N-term
truncation of (1).
(fA/fa) = constant for any N.
3. The sequence of approximants converges for increasing N.
Choosing an approximant that satisﬁes the above deﬁnition will lead to a
uniformly convergent sequence as Nis increased that preserves the correct
asymptotic behavior. Note that in the above deﬁnition, faneed not be exact;
in fact, only the leading order is typically needed to construct an adequate
approximant. We seek approximants whose unknown coeﬃcients can be gen-
erated with ease. We do not wish for the construction of approximants to
undergo more ﬂoating-point operations than performed in an accurate nu-
merical solution. Most of the approximant coeﬃcients calculated here may be
obtained recursively using a few simple series relations listed in Appendix A.
This is in contrast with Pad´es and their extensions, which often require the
inversion of a linear system [3, 4, 10]. As shall be apparent, a theme of this
work is to ﬁnd/impose the simplest asymptotic approximant form possible
while maintaining the desired accuracy and precision.
Approximants are closed form functions that may be analytically diﬀeren-
tiated to obtain auxiliary properties without the loss of accuracy that occurs
with discretized solutions. Also, physically relevant properties may be cast
as unknowns within an approximant, and the approximant can be used as a
predictor for such properties [19, 16, 17, 18]. This feature of approximants
provides a signiﬁcant problem-solving advantage.
The paper is organized as follows. In Section 2, we review recent ap-
plications of asymptotic approximants towards the truncated virial series
of thermodynamics. The analysis is recast here as part of a new uniﬁed
framework that only requires the use of a few simple series relations. In
Section 3, an algorithm is provided to construct asymptotic approximants
for problems in general. The method is then applied to three nonlinear
boundary value problems in Section 4. In Section 4.1, the Sakiadis boundary
layer problem is solved using two types of asymptotic approximants - one
which is both accurate and simple in form, and another which matches its
higher-order asymptotic behavior and is capable of predicting the wall-shear
coeﬃcient and other properties to a higher precision than reported before.
In Section 4.2, the Blasius boundary layer problem is solved using a sim-
ple asymptotic approximant, which is also capable of predicting important
quantities such as the wall-shear coeﬃcient. In Section 4.3, an asymptotic
approximant is used to solve the Flierl-Petviashvili monopole problem and
predict relevant properties, which are shown to agree with newly generated
numerical results. Key ﬁndings are summarized in Section 5.
2 Relevant background and results from virial-
In this section, we review the recent application of asymptotic approximants
towards problems in thermodynamics. In doing the review, simpliﬁcations
and details not previously disclosed are elucidated here and also incorporated
into the algorithm of Section 3.
The virial series is an equation of state formulated as a series expansion
about the ideal gas limit  (ρ→0), and is expressed as
Bn(T)ρn, B1= 1 (3)
where Pis the pressure, ρis the number density, kis the Boltzmann constant,
and Tis the temperature. The virial coeﬃcients Bnare functions of Tas
well as other physical parameters describing a given intermolecular potential.
Speciﬁcally, the nth virial coeﬃcient is deﬁned as an integral over the position
of nmolecules . In the absence of an exact equation of state, the virial
series represents ﬂuid properties at low density more accurately than any
other theoretical or computational method, since it is eﬀectively the Taylor
series of the exact solution about ρ=0. A barrier to its usage is that the
number of integrals appearing for each viral coeﬃcient increases rapidly and
nonlinearly with the order of the coeﬃcient. Consequently, signiﬁcant eﬀort is
spent to develop eﬃcient algorithms to compute higher-order coeﬃcients [22,
23]. As an indication of the diﬃculty involved, only twelve virial coeﬃcients
are currently known (to within some precision) for the hard-sphere model
ﬂuid . For more realistic model ﬂuids, signiﬁcantly fewer coeﬃcients are
Even with several coeﬃcients, the virial series often converges only in a
small region near ρ= 0. Pad´e approximants have been applied to extend
this region, with a common example being the Carnahan-Starling equation of
state for hard-sphere ﬂuids . With only a ﬁnite number of terms available,
the choice of approximant (be it Pad´e or not), is important as convergence
may be accelerated by choosing an approximant with consistent asymptotic
behavior following Deﬁnition 1.1. In the following subsections, we review the
recent application of asymptotic approximants towards divergent and slowly
converging truncated virial series to construct accurate isotherms and predict
2.1 Soft-sphere ﬂuids
To our knowledge, the ﬁrst application of a non-Pad´e asymptotic approxi-
mant towards virial series is given in  and is used to describe isotherms
for soft-sphere ﬂuids. This problem highlights the key steps in formulating
an asymptotic approximant used later in this paper, and we thus review the
technique carefully. Soft-sphere ﬂuids  are deﬁned by a molecular poten-
tial that behaves like r−h, where ris the distance between molecules and h
is a “hardness” parameter. In the limit as h→ ∞, the soft-sphere molecular
potential limits to that of the hard-sphere. For soft spheres, it is typical to
rewrite (3) in terms of reduced virial coeﬃcients ¯
Bnand reduced density ˜ρ
using the scalings provided in . Equation (3) becomes
ρkT = 1 +
where Zis the compressibility factor and the coeﬃcients ¯
Bnare now func-
tions of the hardness h. The above series diverges (see Fig. 1 for h= 4),
with a radius of convergence that decreases with decreasing h. While (4)
accurately describes the low-density behavior, an approximant can be con-
structed by bridging (4) with the large density limit for soft-spheres:
Z∼C˜ρh/3as ˜ρ→ ∞.(5)
Note that if h/3 is not an integer, it is impossible to capture the above
behavior with a standard Pad´e. However, an asymptotic approximant such
ZA= 1 +
limits to (5) for any hand allows for an easy evaluation of its coeﬃcients, such
that the Taylor expansion of (6a) about ˜ρ= 0 is exactly (4). An expression
for the unknown coeﬃcients Anis found by equating the original series (4)
with the approximant form (6a) and then solving for the series in An:
= 1 +
It is now clear that the Ancoeﬃcients are the coeﬃcients of the Taylor ex-
pansion of the left-hand side of the above expression about ˜ρ= 0. Such coef-
ﬁcients are easily obtained using J. C. P. Miller’s formula  for recursively
evaluating the expansion of a series raised to any power; see equation (29) in
Appendix A. The recursion for the coeﬃcients is
Bj+1An−j, A0= 1.(6b)
Together, (6a) and (6b) provide an asymptotic approximant which converges
for all ˜ρin the ﬂuid regime, as shown in  and Fig. 1 (for h=4). Note
that (6b) reduces to the formulae for A2through A10 given in the appendix
of , and can be used to generate higher-order approximants as higher-
order virial coeﬃcients become available. The same procedure used above
will be applied in Section 4 to obtain asymptotic approximants for nonlinear
boundary value problems of mathematical physics.
Figure 1: Compressibility factor Zversus reduced density ˜ρfor the r−4soft-
sphere ﬂuid. A comparison is shown between the N-term virial series (3)
(−−) labeled as SN, the corresponding approximant (6) (−) labeled as AN,
and data from molecular simulation  (•). The plot spans the entire ﬂuid
regime, as the last data point is where freezing occurs. Virial coeﬃcients
used to generate the curves are taken from .
2.2 Prediction of Critical Properties
An important property of asymptotic approximants is their ability to predict
unknown quantities. Pad´e and other approximants have long been used to
predict critical properties of lattice models [28, 29, 6, 7]. With the advent of
asymptotic approximants, this approach has recently yielded accurate pre-
dictions of critical properties of model ﬂuids [17, 18]. The prediction begins
by constructing the approximant in the same manner as the previous section.
The virial series describing P(ρ) along the critical isotherm (i.e. using
coeﬃcients evaluated at the critical temperature T=Tc) captures low density
behavior but diverges due to a branch point singularity at the critical density
ρc. The behavior of ﬂuids near the critical point obeys a universal scaling
, T =Tc, ρ →ρc,(7)
where Pcis the critical pressure, Dis a critical amplitude, and (for real ﬂuids)
δis a positive non-integer critical exponent. The knowledge of a low density
series as well as an asymptotic behavior at higher density are the necessary
ingredients for the application of an asymptotic approximant. In particular,
the series (3) is accurate in a region where (7) does not apply and inaccurate
in the region where (7) is accurate. An approximant is provided in  that
captures both regions (low density and the critical region); it is given by
Following the same approach of the previous subsection, one may equate (8a)
with the virial series (3) evaluated at T=Tcand solve for the series in An:
where it becomes clear that the Ancoeﬃcients are, in fact, those in the Taylor
expansion about ρ= 0 of the left-hand side of the above expression. The An
coeﬃcients are obtained by replacing (1 −ρ/ρc)−δwith its expansion about
ρ= 0 and then taking the Cauchy product (equation (27) in Appendix A)
of the two series in the left-hand side of the above expression. This leads to
Equation 4b of :
If ρc,Tc, and Pcare known, one then inserts (8b) into (8a) to construct
the approximant, which captures both the low-density and critical region,
as shown in . However, it is typically the case that critical properties
are poorly known and diﬃcult to compute through molecular simulation.
Fortunately, the approximant may be used to predict these quantities. For
instance, if ρcis unknown and all other parameters are known, one may
sacriﬁce the highest unknown coeﬃcient ANand instead predict ρc. This is
equivalent to setting AN=0 in (8b), leading to Equation 7 of ,
c= 0 (8c)
which is a polynomial in ρc. Although this predicts multiple roots for ρc,
complex or negative values may be eliminated. As one increases Nand tracks
the roots, typically one set of positive roots will converge to a limit point more
rapidly than others, as shown in Fig. 2a for the Lennard-Jones model ﬂuid.
Taking the fastest converging ρcsequence in the ﬁgure and substituting it
into (8b) and (8a) for each Nleads to a set of approximant isotherms, shown
in Fig. 2b. Note in the ﬁgure that the approximant converges more rapidly
and uniformly than the virial series as Nis increased.
0 0.2 0.4 0.6 0.8 1 1.2
Figure 2: (a) Predictions of the critical density from (8c) for the Lennard-
Jones ﬂuid. A dashed line indicates the value predicted from Monte-Carlo
simulations . (b) Critical isotherm of the Lennard-Jones ﬂuid. A compar-
ison is shown between the N-term virial series (3) (−−) labeled as SNand
the corresponding approximant (11) (−) labeled as AN. The most rapidly
converging branch of ρcroots (lowermost points) in (a) is used as an input
to the approximants shown in (b). Virial coeﬃcients used to generate the
plots are taken from .
Now if both ρcand Pcare unknown, one may sacriﬁce the two highest
coeﬃcients (i.e. AN=AN−1= 0), leading to a system of 2 nonlinear equa-
tions to solve for ρcand Pc. Recently, this approach has been used to predict
properties and construct approximants that describe the entire critical re-
gion  - not just the critical isotherm. Results in Section 4 demonstrate
that this method is also useful in the prediction of quantities important to
nonlinear boundary value problems.
As mentioned in the introduction, the virtue of approximants such as (6)
and (8) is the ease with which accurate derivative and integral properties are
generated. For example, the more comprehensive critical approximant given
in  for P(ρ, T ) can be integrated with respect to ρto accurately calculate
the Helmholtz free energy, which may then be diﬀerentiated with respect
to Tto ﬁnd the internal energy and speciﬁc heat. Even if these quantities
are extracted numerically after an approximant is constructed, they may be
resolved on a grid of arbitrary resolution with minimal computational expense
(only memory usage), since the approximant is an analytic expression.
3 General procedure for constructing an asymp-
With the aim of formalizing the method of asymptotic approximants intro-
duced in Section 2, we provide the following algorithm. This procedure is
subsequently applied to power series solutions of nonlinear boundary-value
problems in Section 4.
Algorithm 3.1 Steps for constructing an asymptotic approximant for a prob-
lem whose analytic solution f(x)is elusive:
i. Find a power series solution of the given problem and truncate to N
Deﬁne mas the number of unknowns embedded in the coeﬃcients an;
this value is used in step v below. Note that step i is independent of step
ii and thus their order may be interchanged.
ii. Seek the asymptotic behavior f∼Cfaas x→xa, where xa6=x0and
Cis a content. This may come from either an asymptotic expansion
about xa, a boundary condition, or some independently known limiting
behavior of f. Deﬁne pas the number of unknowns associated with the
asymptotic behavior; this value is used in step v below.
iii. Create an approximant function fAwith N+1 unknowns A0. . . AN. The
function must (a) match the correct asymptotic behavior f∼Cfaas
x→xato at least zeroth order and (b) be generalized to handle arbitrary
N; a function of an “approximant series” is well suited for this:
This is the creative step. The goal is to capture known features of f
while leaving ﬂexibility to continuously merge two (potentially disparate)
iv. Solve for A0. . . ANby imposing the condition that the Nth-order Taylor
expansion of fAabout x0is equal to fS. If g(x) = x, this can be accom-
plished by setting fS=fA, isolating the approximant series within fAto
one side and then Taylor expanding the other side using the relations in
Appendix A. If g(x)6=x, a matrix inversion may be required. In either
case, the resultant expressions for A0. . . ANwill be functions of the m+p
unknowns from steps i and ii.
v. Solve the algebraic system AN= 0, AN−1= 0, . . . AN−m−p+1 = 0 for
the m+punknowns from steps i and ii. Choose the most rapidly con-
verging and physically plausible roots (for example, the physical quantity
the root represents may be real and positive). In this step, unknown co-
eﬃcients in the approximant series are sacriﬁced in order to solve for
other unknowns embedded in these coeﬃcients, while preserving the cor-
rect number of degrees of freedom.
vi. Evaluate all Ancoeﬃcients at the values obtained in step v and substitute
these into the approximant fAfrom step iii.
vii. Ensure that fAconverges as Nis increased. As an additional metric
of the approximant’s ability to approach the correct asymptotic behav-
ior, one may plot fA/faand conﬁrm that, for increasing N, this ratio
converges to a constant as x→xa.
viii. Consistency Checks: Expand fAabout x0to Nth order; if the steps were
executed correctly this will match the ﬁrst Nterms of fS. Take the limit
of fAas x→xaand conﬁrm that it agrees with fato zeroth order.
Finally, if available, verify that the approximant agrees with a full nu-
merical solution of the problem.
4 Application to nonlinear boundary value
Asympotic approximants are now applied to three nonlinear boundary value
problems. In Section 4.1, we ﬁnd approximant solutions to the Sakiadis
boundary layer problem, where we are able to take advantage of the full
asymptotic behavior, enabling us to obtain new benchmark values. In Sec-
tion 4.2, an approximant is constructed for the Blasius boundary layer prob-
lem using the leading order asymptotic behavior; this approximant conﬁrms
existing numerically-obtained benchmark values. In Section 4.3, an approx-
imant for the Flierl-Petviashvili monopole problem is constructed using the
leading order asymptotic behavior. In this application, newly obtained nu-
merical results are conﬁrmed and a new asymptotic constant is explored.
4.1 The Sakiadis problem
The Sakiadis problem describes steady-state developing ﬂow created by a
moving plate in an otherwise stagnant ﬂuid . It shares the same bound-
ary layer governing equation as that of the Blasius problem (discussed in
Section 4.2), but with diﬀerent boundary conditions. As shown in , a sim-
ilarity transform of these equations leads to the following nonlinear boundary
value problem in f(η):
2f000 +ff00 = 0 (9a)
f(0) = 0, f0(0) = 1, f 0(∞) = 0.(9b)
As shown by Blasius , substituting the solution form
into (9a) leads to the following recursion for the series coeﬃcients
(j+ 1)(j+ 2) aj+2 an−j
2(n+ 1)(n+ 2)(n+ 3) .(10b)
The recursion above requires knowledge of the ﬁrst three coeﬃcients in order
to generate the full series. The ﬁrst two coeﬃcients a0=f(0) and a1=f0(0)
are given by the ﬁrst two boundary conditions in (9b). The third coeﬃcient
is prescribed by the “wall shear” parameter
and is typically obtained from a numerical solution of (9) (see [35, 36, 37]
for example). Recently, a semi-analytical ﬁxed point method was used to
estimate κ; see  for details and a comprehensive review and comparison
of previous numerical predictions. Here, asymptotic approximants will be
used to predict κ.
From the inﬁnite condition in (9b), it follows that the solution limits to
a constant Cat large η:
η→∞ f≡C. (11)
Higher order asymptotic methods provided in Appendix B.1 enable the iden-
tiﬁcation of an additional asymptotic constant, G, as:
η→∞ (f−C)eηC/2≡G. (12)
The constants G,C, and κwill be predicted using asymptotic approximants
in the following subsections.
4.1.1 Simple asymptotic approximant
First, we consider a simple approximant of the form
fA=C−C 1 +
which automatically satisﬁes both (9b) and (11). The assumed form (13a) has
the appearance of a Pad´e, but in fact it is not. Since Cis an unknown to be
predicted by the approximant, the above cannot be considered a Pad´e for the
quantity f−C. Also, if the terms in (13a) were combined through a common
denominator, a speciﬁc relationship between the numerator and denominator
coeﬃcients would be obtained. Thus (13a) cannot be considered a Pad´e for f.
We shall see below that the approximant given by (13a) contains coeﬃcients
which are, in fact, easier to compute than those of a standard Pad´e.
The coeﬃcients A0. . . AN,C, and κare calculated such that the N-
term Taylor expansion of (13a) about η= 0 is exactly equal to the N-term
truncation of (10). Equating (13a) with (10) and re-arranging leads to the
= 1 +
Expanding the left-hand side above (utilizing equation (28) in Appendix A)
provides the following recursion for the coeﬃcients, which are now functions
of κand C:
ajAn−j, A0= 1.(13b)
We now sacriﬁce the coeﬃcients ANand AN−1to simultaneously predict κ
and C. Setting AN=AN−1= 0 in (13b), we arrive at
Once the preceding coeﬃcients are obtained from (13b) and substituted into
the above, (13c) becomes a system of nonlinear equations in κand C, and
may be solved algebraically. The most rapidly converging κand Croots
of (13c) are given in Table 1 up to an optimal asymptotic truncation of
N= 11. That is, for N > 11, |κN+1 −κN|no longer consistently decreases
as Nis increased.
We also record the value Sas the magnitude of the singularity, ηs, closest
to η=0 in the complex ηplane for the Sakiadis problem, as predicted by (13)
and listed in Table (1). This value limits the radius of convergence of the
Lastly, Cand κare substituted into (13b) for all non-zero coeﬃcients
(A1through AN−2) to construct the approximant series in (13a). The ap-
proximant (13) is compared with the series and numerical solution of (9)
in Fig. 3. Although we are using a simple approximant form that enables
correspondingly straightforward coeﬃcient generation, the approximant con-
verges surprisingly well, as can be seen in Fig. 3a. In Fig. 3b, we conﬁrm
that f0and f00 are accurately obtained from the approximant, once fhas
reasonably converged (here, for N= 11). Unfortunately, our estimates for κ
and Ccease to converge for N > 11, which limits our ability to predict the
asymptotic constant Ggiven by (12). This can been seen in a plot of the
Geﬀ = (f−C)eηC/2, Geﬀ →Gas η→ ∞ (14)
versus η, which, when computed by the approximant (13), only converges
in a small region near η= 0 as shown in Fig. 3b. The reason for this poor
convergence behavior is that the approximant form (13) does not incorporate
the asymptotic correction behavior that leads to (12), and thus Geﬀ cannot
Table 1: Predictions from approximants for the Sakiadis problem. For com-
parison, the numerical values of κand Cin the bottom row were computed
using the Chebfun  package to solve (9) via rectangular collocation with
η=∞replaced with a ﬁnite surrogate η∞given in the table. Note that
the recent κestimates given in [40, 38] (reported to 7 digits) agree with the
ﬁrst 7 digits of our numerical predictions and predictions from (15). The nu-
merical values of Swere obtained by inserting the numerical κand Cvalues
into (13), locating the singularity of smallest magnitude, and systematically
increasing the approximant order (up to N= 200), where the value of Shas
converged to within the reported digits.
5 -0.3879 2.1607 3.76 -0.464241808775 1.57791591603 -1.9379232426
7 -0.4341 1.7275 3.98 -0.445957694266 1.61001774844 -2.0928590899
9 -0.4421 1.6418 4.05 -0.443970941603 1.61533742596 -2.1255327825
11 -0.4430 1.6284 4.07 -0.443769095918 1.61603802403 -2.1306162759
13 − − − -0.443750163304 1.61611659263 -2.1312640887
15 − − − -0.443748473247 1.61612459984 -2.1313373822
17 − − − -0.443748326909 1.61612536888 -2.1313450758
19 − − − -0.443748314499 1.61612543984 -2.1313458428
21 − − − -0.443748313462 1.61612544619 -2.1313459165
23 − − − -0.443748313376 1.61612544675 -2.1313459234
25 − − − -0.443748313368 1.61612544681 -2.1313459241
26 − − − -0.443748313384 1.61612544669 -2.1313459226
27 − − − -0.443748313370 1.61612544680 -2.1313459239
28 − − − -0.443748313369 1.61612544680 -2.1313459240
29 − − − -0.443748313369 1.61612544681 -2.1313459241
30 − − − -0.443748313369 1.61612544681 -2.1313459241
numerical κ C S
from (13) η∞
shooting  -0.44374733 1.61611200 4.07217 20
quadrature  -0.44374831 1.61612518 4.07217 23.20512
collocation -0.44374831 1.61612545 4.07217 30
0 5 10 15 20
0 5 10 15 20
Figure 3: (a) Comparison between the N-term series solution (10) (−−)
labeled as SN, the corresponding approximant (13) (−) labeled as AN, and
the numerical solution  (•) to the Sakiadis problem (9). (b) Derivatives
of approximant (13) with N=11 and Geﬀ given by (14) compared with the
numerical solution  (•).
4.1.2 Asymptotic approximant with exponential corrections
We now consider an approximant,
that not only matches the zeroth order inﬁnite behavior, but also the form
of the exponential corrections from the asymptotic expansion of fas η→ ∞
provided in Appendix B.1. Note that A1in the series above is the unknown
asymptotic constant Gdeﬁned in (12). After expanding (15a) about η= 0
and matching like terms with those of the series solution (10), we arrive at
a system of Nlinear equations,
where the coeﬃcient matrix is a Vandermonde matrix with a known explicit
formula for its inverse . Once the system (15b) is solved, the Anco-
eﬃcients become functions of the unknowns Cand κ. Like the previous
approximant, we sacriﬁce the last two coeﬃcients and solve the nonlinear
AN(κ, C) = 0, AN−1(κ, C) = 0 (15c)
to obtain values for Cand κ, provided in the 5th and 6th columns of Table 1.
In the table, we report values up to N= 30, after which the 13th digit
oscillates without converging to a discernible limit point. The converged
obtained from approximant (15) agree with numerical predictions, and pro-
vide 5 additional digits of precision. Also, for the ﬁrst time, a value of the
is reported; convergence to this value is shown in the rightmost column of
Table 1, obtained by substituting the predictions for Cand κinto the ex-
pression for A1. Once Cand κare substituted into the expressions (arising
from the solution to (15b)) for all non-zero coeﬃcients (A1through AN−2),
the approximant series in (15a) may be constructed.
The approximant (15) is compared with the series and numerical solu-
tion of (9) in Fig. 4. The new approximant converges more rapidly than the
simpler approximant (see Fig. 3 for comparison). Also, the new approximant
captures the exponential corrections inasmuch as Geﬀ converges into the large
ηasymptotic regime and approaches G, as shown in Fig. 4b. This behav-
ior is expected, as the correct asymptotic behavior has been incorportated
explicitly into the approximant and convergence should be uniform for large
For the Sakiadis problem, a relatively simple approximant that meets the
inﬁnite condition in (9b), such as approximant (13), is adequate for describing
fand its derivatives. However, in comparing the Geﬀ curves of Figs. 3b and
4b, it is clear that an approximant such as (15) which includes asymptotic
corrections, is needed to predict G. As we shall see in the next section, simple
approximants can sometimes also be used to predict higher-order asymptotic
0 5 10 15 20
S11 A7, 9, 11
0 5 10 15 20
Figure 4: (a) Comparison between the N-term series solution (10) (−−)
labeled as SN, the corresponding approximant (15) (−) labeled as AN, and
the numerical solution  (•) to the Sakiadis problem (9). (b) Derivatives
of approximant (15) with N=7 and Geﬀ given by (14) compared with the
numerical solution  (•).
properties, provided that the approximant converges at large enough ηfor
the asymptotic behavior to be reached.
Lastly, it is interesting to note that approximant (15) contains no singu-
larities, and thus cannot be used to predict the closest singularity ηs(and
S) directly, as was done in Section 4.1.1 for approximant (13). However,
if we use either the new high precision values of κand Cfrom approxi-
mant (15) or the numerical values listed in the table as inputs to approxi-
matnt (13), the location of the singularity converges to the within the indi-
cated digits to ηs=−1.211393 + 3.88781i and thus a radius of convergence
of S≡ |ηs|= 4.07217.
Although the implementation is diﬀerent, the approximant (15) is similar
in style to that of the approximant of the Kidder boundary value problem
given recently in , where an exponential decay of the derivative at inﬁnity
is incorporated by changing the independent variable of a Pad´e to reﬂect
this behavior. If convergence of this approximant for increasing order were
considered, this approach would fall within the framework of the asymptotic
approximants presented here.
The errors in the approximants of this section are given in Table 2 to en-
able comparison with other approximate treatments of the Sakiadis problem;
see for example .
Table 2: The inﬁnity norm of the error in the Sakiadis approximants and
their 2nd derivatives (to within 1 signiﬁcant digit), deﬁned respectively as
numerical||∞on the interval 0 ≤η≤20.
13 − − 9×10−62×10−6
15 − − 8×10−72×10−7
20 − − 2×10−93×10−10
25 − − 1×10−11 5×10−12
30 − − 1×10−11 4×10−13
4.2 The Blasius problem
The Blasius problem is the archetypal boundary-layer problem found in most
undergraduate ﬂuid mechanics books, describing the boundary layer due to
a moving ﬂuid over a stationary ﬂat plate . The diﬀerential equation
in f(η) is the same as the Sakiadis problem (and arises after a similarity
transform is applied to the governing equations),
2f000 +ff00 = 0,(16a)
while the boundary conditions are now
f(0) = 0, f0(0) = 0, f 0(∞) = 1.(16b)
The solution to (16) may be constructed as a power series (10) as in the
Sakiadis problem. However, one diﬀerence is that the Blasius series solution
skips every two terms, starting with the ﬁrst two, which are zero as imposed
by the boundary conditions at η=0 in (16b). Again, the wall shear parameter
is deﬁned as
which, along with the asymptotic properties 
η→∞ exp η2
will here be predicted by the asymptotic approximant, in a similar fashion as
done by  (for κusing Pad´es). Recent numerical and analytical predictions
of these quantities, including a review of previous results may be found in [44,
38, 45, 46].
The asymptotic behavior of the Blasius problem as η→ ∞ is given as
(see Appendix B.2)
(η+B)2[1 + O(1
(η+B)2)],as η→ ∞.(18)
For comparison, the exponential correction analogous to (18) for the Sakiadis
problem is (36) in Appendix B.1. While we were able to incorporate (12)
directly in the Sakiadis approximant (15), this is not an option for the Blasius
problem since the asymptotic form (18) contains a singularity at η=−Band
it is known that B < 0 . To clarify this issue, note that (18) is fully valid
as η→ ∞. However, our methodology for constructing an approximant relies
on the uniﬁcation of asymptotic limits such that the ﬁnal form is capable of
describing all η > 0, and here is where (18) poses an issue in its ability to
inform such an approach.
While we cannot directly use the asymptotic expansion for the Blasius
problem, we can use a simple approximant that agrees asymptotically to 1st
order (i.e. with (16b) and (17a)) and then verify that it approaches the
exponential correction (17b) as η→ ∞. We use an approximant of the form
fA=η+B−B 1 +
which automatically satisﬁes both (16b) and (17a). As similarly discussed in
the context of the Sakiadis Approximant (13a), the above form is not a Pad´e
approximant. Firstly, Bis an unknown parameter in the approximant, and
so (19a) cannot be considered a Pad´e for f−η−B. Secondly, if one combines
the terms of (19a) through a common denominator, the coeﬃcients of the
numerator will have an explicit dependence on those in the denominator,
whereas this is not the case for standard Pad´es.
The coeﬃcients A0. . . AN,B, and κare now calculated such that the N-
term Taylor expansion of (19a) about η= 0 is exactly equal to the N-term
truncation of (10). Following the same procedure as in Section 4.1.1, we
arrive at the following recursion for the coeﬃcients, which are now functions
of κand B:
˜ajAn−j, A0= 1,(19b)
where ˜a1=−1 and ˜aj>1=aj. The coeﬃcients ANand AN−1are now
sacriﬁced to simultaneously predict κand B. Setting
AN(κ, B) = 0, AN−1(κ, B) = 0 (19c)
in (19b) leads to two nonlinear equations, whose most rapidly converging κ
and Broots are given in Table 3, along with the magnitude of the singularity
closest to η= 0 in the approximant, denoted by S. The table reports values
up to N= 50, beyond which convergence cannot be established in the 8th
digit. Our predictions converge to within 7 digits of the benchmark value
for κ, 6 digits for B, and 2 digits for the radius of convergence S. Again, we
determine Sas the magnitude of the closest singularity, ηs, from the origin in
the complex ηplane. Although not shown in the table, the approximant (19)
is consistent with the literature  in that, as Nis increased, it predicts ηs
to lie on the negative real axis.
Once Band κare substituted into (19b) for all non-zero coeﬃcients (A1
through AN−2), the approximant series in (19) may be constructed. The
approximant (19) is compared with the series and numerical solution of (16)
in Fig. 5. Like the simple Sakiadis approximant given in Section (4.1.1),
this basic Blasius approximant converges surprisingly well, as can be seen
in Fig. 5a. In Fig. 5b, we conﬁrm that f0and f00 are accurately obtained
from the approximant, once fhas reasonably converged (here, for N= 30).
Unlike the simple Sakiadis approximant, the Blasius approximant is capable
of picking up a higher-order asymptotic quantity - namely Qdeﬁned in (17b).
This is seen in Fig. 5c, where an eﬀective Q, deﬁned as
Qeﬀ = exp η2
2f00, Qeﬀ →Qas η→ ∞ (20)
is plotted versus ηfor various series and approximant order. While the ap-
proximant converges directly to a constant Qvalue at large η, it is striking
that the series solution also reaches the Qplateau prior to diverging. This
may be a result of the Blasius series having a region of convergence that over-
laps with asymptotic eﬀects felt at relatively small η. As seen in Fig. 5c, both
Table 3: Predictions from the Blasius approximant (19). The inﬁnity norm
of the error in the approximant and its 2nd derivative are recorded below (to
within 1 signiﬁcant digit), deﬁned respectively as E=||fA−fnumerical||∞and
numerical||∞on the interval 0 ≤η≤8.8.
N κ B S E E2
5 0.12545065 -6.08950239 9.85302 3×1002×10−1
10 0.30018461 -2.00003122 4.47053 3×10−13×10−2
15 0.32626056 -1.77168204 4.59756 5×10−25×10−3
20 0.33090243 -1.73173129 4.92273 1×10−21×10−3
25 0.33181978 -1.72323142 5.18569 2×10−32×10−4
30 0.33200793 -1.72133641 5.43362 5×10−45×10−5
35 0.33204717 -1.72090879 5.64460 1×10−41×10−5
40 0.33205518 -1.72081494 5.69196 3×10−52×10−6
45 0.33205693 -1.72079309 5.68929 5×10−64×10−7
50 0.33205731 -1.72078801 5.68933 3×10−73×10−8
numerical [45, 47] 0.332057336215196 -1.7207876575205 5.6900380545 0 0
the approximant and the numerical solution converge to Q≈0.1115, which
is consistent with the value Q=0.111483755 obtained numerically by  and
semi-analytically (i.e. dependent on numerical κ) in 1; in both references,
the value is reported well-beyond these digits, but both round to the value
above. Since the behavior (20) is not explicitly incorporated into the approx-
imant (19), the convergence of Qeﬀ is not uniform at large η. For example,
in Fig. 5c, the behavior of the N= 20,30,40 approximants (labeled A20,
A30, A40) demonstrates the non-uniform convergence that will occur for any
order Nat some value of η.
The errors in the approximant are given in the last two columns of Ta-
ble 3 to enable comparison with other approximate treatments of the Blasius
problem [19, 43, 44, 38, 46].
The virtue of approximant (19) is that it is globally accurate from small
to large η, is able to predict κand B(i.e. does not rely on these values as
inputs), and has coeﬃcients that are generated from a simple recursion. A
recent alternative globally accurate approximation to the Blasius solution is
provided in . An attractive feature of this alternative approximation is
that it has a simpler form than the asymptotic approximant (19). However,
1Qin  is deﬁned as e−B2/4multiplied by the Qdeﬁned here.
A20, 30, 40, 50
S20, 40, 50 S10
Figure 5: (a) Comparison between the N-term series solution (10) (−−)
labeled as SNand the corresponding approximant (19) (−) labeled as AN
to the Blasius problem (16). (b) Derivatives of approximant (19) with N=30.
(c) Qeﬀ given by (20) as predicted by the series (10) and approximant (19).
For comparison, the numerical solution from  is shown in all above ﬁgures
the form in  requires that κis known beforehand. Nevertheless, one may
directly compare the error in f00
Agiven in Table 3 with that given in ﬁgure 2
4.3 The Flierl-Petviashvili problem
The Flierl-Petviashvili (FP) equation in similarity variables u(r) is used to
describe vortex solitons in the ocean, atmosphere , and Jupiter’s red
spot , and is given by
ru0−u−u2= 0 (21a)
u0(0) = 0, u(∞)=0.(21b)
The coeﬃcients of the power series solution
to (21) are given by
which requires the speciﬁcation of the ﬁrst two coeﬃcients to generate the
remaining even coeﬃcients; all remaining odd terms are zero. The coeﬃcient
a1=0 is known from the boundary conditions (21b) and a0is an unknown
which, along with the asymptotic property
will be predicted by the approximant, in a similar fashion as done by [19, 50]
(for zusing Pad´es). The property (23) above follows from an asymptotic
expansion of the FP problem as r→ ∞ (derived in Appendix B.3):
√r[1 + O(1
r)],as r→ ∞.(24)
As was the case for the Blasius problem, a singularity in the asymptotic form
(here at r=0 in the above) prevents direct incorporation of (24) to construct
an approximant. However, like the Blasius approximant of Section 4.2, we
shall again use a simple approximant that agrees asymptotically to zeroth
order (i.e. with the 2nd condition of (21b)) and then verify that it approaches
the correction (24) as r→ ∞.
We consider a simple approximant of the form
which automatically satisﬁes both boundary conditions of (21), including
the condition u(∞) = 0 not explicitly captured by (22). Note that (25a) is a
Pad´e. The simple structure of this Pad´e, however, allows for the unknowns
to be calculated without inverting a matrix. The coeﬃcients A0. . . ANand z
in (25a) are now calculated such that the N-term Taylor expansion of (25a)
about r= 0 is exactly equal to the N-term truncation of (22). Following the
same procedure as in Section 4.1.1, we arrive at the following recursion for
the coeﬃcients, which are now functions of z:
ajAn−j, A0= 1.(25b)
The coeﬃcients above mimic the series (22) in that it skips odd coeﬃcients.
The unknown zin (25b) is determined by sacriﬁcing the degree of freedom
normally used to compute AN. This is equivalent to setting AN= 0 in (25b),
Note that since the ajcoeﬃcients are polynomials in z, (25c) is also a polyno-
mial in z. The most rapidly converging z-roots of (25c) (for increasing N) are
given in Table 4. Once zis substituted into (25b) for all non-zero coeﬃcients
(A1through AN−1), the approximant series in (25a) may be constructed.
The magnitude of the singularity closest to r= 0 in the approximant (25) is
also given in Table 4, denoted by S. Although not shown in the table, the
approximant (25a) predicts that, within the reported digits, the singularity
appears to lie on the positive imaginary axis.
The approximant (25) is compared with the series and numerical solution
of (21) in Fig. 6. As for the simple Sakiadis approximant given in Sec-
tion 4.1.1 and the Blasius approximant of Section 4.2, the FP approximant
also converges uniformly, as can be seen in Fig. 6a. In Fig. 6b, we conﬁrm
that u0is accurately obtained from the approximant once uhas reasonably
Table 4: Predictions from the Flierl-Petviashvili approximant (25a). The
numerical method for computing the values in the bottom row is described
in Appendix C. The inﬁnity norm of the error in the approximant is also
recorded below (to within 1 signiﬁcant digit), deﬁned as E=||uA−unumerical||∞
on the interval 0 ≤r≤10.
N z S E
4 -1.5 2.82843 9×10−1
6 -2.14039 2.57284 3×10−1
8 -2.34792 2.47227 4×10−2
10 -2.38564 2.60878 6×10−3
12 -2.39117 2.59927 7×10−4
14 -2.39185 2.61160 2×10−4
16 -2.39196 2.61154 1×10−5
numerical -2.3919564032 2.611541077 0
converged (here, for N= 12). Like the Blasius approximant, the FP approx-
imant is capable of picking up a higher-order asymptotic quantity - here, D
deﬁned in (23). This is seen in Fig. 6b, where an eﬀective D, deﬁned as
Deﬀ =uer√r, Deﬀ →Das r→ ∞ (26)
is plotted versus rfor various approximant order. As Nis increased, the
approximant Deﬀ appears to be converging to the numerical solution, which
is in turn converging to a value of
which may serve as a useful metric for future analytical and numerical solu-
tions of the FP problem. Again, Fig. 6 shows that convergence in Deﬀ is not
uniform at large r, as expected by the assumed form of the approximant.
The errors in the approximant are given in the rightmost column of Ta-
ble 4 to enable comparison with other approximate treatments of the FP
equation; see for example .
In this work, we formalize a new approach to sum series. We provide both a
methodology and examples that demonstrate how asymptotic approximants
Figure 6: (a) Comparison between the N-term series solution (22) (−−)
labeled as SNand the corresponding approximant (25) (−) labeled as AN
to the Flierl-Petviashvilli problem (21). (b) Derivatives of approximant (25)
with N=12 and Deﬀ given by (26). For comparison, the numerical solution
(described in Appendix C) is shown in all above ﬁgures as •s.
may be constructed. The key feature is that such approximants are designed
so that the asymptotic behaviors in two regions of the domain may be joined.
This approach has seen recent success in the analytic continuation of “virial”
series that describe the pressure-density-temperature dependence of various
model ﬂuids, where no diﬀerential equation is available and usually only the
ﬁrst 3 to 12 terms of these (often divergent) power series are known [16, 17,
Here, we provide an additional application for asymptotically consistent
approximants - namely, a remedy for divergent power series that arise as
solutions to nonlinear ordinary diﬀerential equations. The analytic forms
provided here enable symbolic diﬀerentiation and thus allow analytical eval-
uation of the ﬂow ﬁeld (shear stress, vorticity, etc.) to any desired resolution
at low computational expense. Additionally, the approximant coeﬃcients are
relatively straightforward to obtain, as they often only require the implemen-
tation of a few series identities, enabling them to be generated typically via
a simple recurrence relation.
Note that improvements to the forms given here are easily made, de-
pending on how many parameters are to be embedded in (or predicted by)
the approximant. Here, we use the approximant to not only ﬁnd a solution,
but also provide the as-of-yet most accurate and precise values of the wall
shear, asymptotic constants, and singularity of smallest magnitude for the
Sakiadis boundary layer. If better estimates become available from numer-
ical (or other) techniques, these may be used as inputs to the approximant
to improve global accuracy. For the Blasius problem and Flierl-Petviashvilli
problem, more accurate values for some of these properties are, in fact, avail-
able and could have been used as inputs to the approximants; we chose
instead to predict these quantities to demonstrate that the method is self
contained and does not rely on external data.
The results presented here motivate the development and application of
asymptotic approximants to other problems where divergent, truncated, and
underspeciﬁed series arise. Considering the capabilities aﬀorded and the ease
with which asymptotic approximants are generated, they deserve considera-
tion as an alternative to the more traditional Pad´e approximants in problems
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A Useful Series Formulae
The following relations may be used to develop a recursion for the coeﬃcients
of an asymptotic approximant, allowing one to avoid solving an algebraic
system. The ﬁrst relation is the well-known Cauchy product of two series :
Setting both sides of (27) equal to one and rearranging, the recursion leads
to a representation for the N-term expansion of the reciprocal of a series:
The generalization of (28) for the N-term expansion of a series raised to any
real power sis given by J. C. P. Miller’s formula :
(js −n+j)ajAn−j, A0= (a0)s.(29b)
B Asymptotic Expansions
B.1 Sakiadis Problem
We wish to determine the η→ ∞ asymptotic behavior of the solution f(η)
to the Sakiadis equation (9). For purposes of the analysis, the inﬁnite con-
straint (11) alone suﬃces. We write:
f=C+h(η),with h→0 as η→ ∞ (30)
where Cis a constant and h(η) is a function to be determined. The form (30)
is substituted into equation (9a) to obtain
2h000 + (C+h)h00 = 0.(31)
The equation (31) may be simpliﬁed by noting that terms quadratic in hare
small relative to other terms as η→ ∞, i.e.:
2h000 +Ch00 = 0,as η→ ∞.(32)
The solution of this equation is
h∼Ge−Cη/2,as η→ ∞,(33)
where Gis a constant. Higher order corrections can be obtained by assuming
h∼Ge−Cη/2+D(η),as η→ ∞,(34)
and applying the method of dominant balance  when substituted into (31),
which leads to
4Ce−Cη ,as η→ ∞.(35)
The asymptotic behavior is written concisely by combining equation (30), (34),
and (35) to yield:
4Ce−Cη +O(e−3C η/2),as η→ ∞.(36)
The above process for obtaining corrections may be repeated and leads to a
series of exponentials with arguments −nCη/2 for n= 1,2,3,4. . . .
B.2 Blasius Problem
We wish to determine the η→ ∞ asymptotic behavior of the solution f(η)
to the Blasius problem (16). For purposes of the analysis, the inﬁnite con-
straint (17a) alone suﬃces. We write:
f=η+B+g(η),with g→0 as η→ ∞,(37)
where Bis an unknown constant and g(η) is a function to be determined.
The form (37) is substituted into (16) to obtain
2g000 + (η+B+g)g00 = 0.(38)
The equation (38) may be simpliﬁed by noting that terms quadratic in gare
small relative to other terms as η→ ∞, i.e.:
2g000 + (η+B)g00 = 0,as η→ ∞.(39)
The solution of this equation is
g00 ∼Qe[−(η2+2Bη)/4] ,as η→ ∞,(40)
where Qis an unknown constant. We can then write:
e[−(ξ+B2)/4]dξ, as η→ ∞ (41)
and after application of integration by parts, we obtain:
g0∼ −2Qe(B2/4) e[−(η+B2)/4]
η+B[1 + O(1
(η+B)2)],as η→ ∞.(42)
The process can then be repeated to ﬁnd an expression for g, i.e.:
ξ+Bdξ, as η→ ∞,(43)
from which integration by parts yields
(η+B)2[1 + O(1
(η+B)2)],as η→ ∞,(44)
which can be combined with the original assumed form (37) to obtain the
(η+B)2[1 + O(1
(η+B)2)],as η→ ∞.(45)
B.3 Flierl-Petviashvili Problem
We wish to determine the r→ ∞ asymptotic behavior of the solution u(r)
to the Flierl-Petviashvili problem (21). For purposes of the analysis, the
inﬁnite constraint in (21b) alone suﬃces. Since u→0 as r→ ∞, it is seen
by inspection of (21) that u2<< u as r→ ∞, leading to the linear equation
r2u00 +ru −r2u= 0,as r→ ∞.(46)
The solution of (46) is a modiﬁed Bessel function of zeroth order (p.
374 equation 9.6.1). The leading-order behavior of this function is given by
equation 9.7.2 (p. 378) of  as
√r[1 + O(1
r)],as r→ ∞,(47)
where Dis an unknown constant.
C Numerical Solution of the Flierl-Petviashvili
The numerical solution to the Flierl-Petviashvili equation used in Table 4 is
obtained by ﬁrst applying the transformation α= 1 −e−rto (21), leading to
dα2−[(1 −α) + 1 + α
dα −u−u2= 0
dα(0) = 0, u(1) = 0.(48)
Note that, like the original FP equation (21), the solution to (48) is not
unique. There is a trivial solution u= 0 and a non-trivial solution - we seek
the latter. The “Shooting Method”  (i.e. iterating on u(0) = zguesses)
is used to eﬀectively convert (48) into an initial value problem, which is
solved via 4th order Runge-Kutta. The boundary condition u(1) = 0, which
informs the shooting iteration, is replaced with u(1 −ε) = 0, where ε << 1.
Table 5 lists numerical predictions of zfor decreasing ε. For each value
of ε, the numerical αstep size used in the Runge-Kutta implementation
was successively decreased until convergence was established to within the
reported digits of the table.
Using the high precision zvalue of -2.3919564032 in Table 5, the mag-
nitude Sof the closest singularity to the origin in the complex r-plane is
computed by applying the ratio-test to (22b) and making a Domb-Sykes plot
of the ratios versus inverse coeﬃcient order , shown in Fig. 7 and taken up
to n= 2000. A linear ﬁt is then made near 1/(n+ 1)=0 with an n=∞inter-
cept of S= 2.611541077, where the digits have been conservatively truncated
since the residual of the linear ﬁt is of order 10−12.
Table 5: Predictions of u(0) ≡zof (48), obtained numerically by shooting
to the boundary condition u(1 −ε) = 0. The reported digits are within
the convergence tolerance of the shooting iteration. For comparison, the
previous benchmark values reported by  and  are z=−2.3919564
and −2.391956403, respectively.
0 0.2 0.4 0.6 0.8 1
0 0.001 0.002
Figure 7: Domb-Sykes ratio plot of the coeﬃcients in (22b) using a0≡z=
−2.3919564032 from Table 5. The intercept at 1/(n+ 1)=0 is 2.611541077.