On the Convergence of Borel Approximants
Donald Lutz
San Diego State University (USA) & Université d'Angers (France)
Algorithms Seminar
October 30, 2000
[summary by Marianne Durand]
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Abstract
For some ``irregular singular'' problems coming from differential
equations, there exist formal power series solutions that are
everywhere divergent. These power series turn out to make sense as
asymptotic expansions of actual solutions. The Borel summation
technique is used to recover convergent representations for these
actual functions solutions.
1 Resummation
Some ``irregular singular'' problems coming from differential
equations have formal power series solutions that are everywhere
divergent. By resummation techniques, one can obtain convergent
solutions [7, 10]. We consider a power series,
solution of a linear differential equation, that is everywhere
divergent, noted
x^{~}(z)=å_{1}^{¥}x_{n}z^{n}.
We assume that it has Gevrey order equal to one, which means
that there exist constants A and c such that
x_{n}£ Ac^{n}n!.
For a function f(z), holomorphic in an angular sector S extending
to infinity and containing the real positive axis, we say that
x^{~}(z) is the Gevrey expansion of order 1 of f(z) if
there exist constants K and C such that

½
½
½
½ 
f(z) 

x_{n}z^{n} 
½
½
½
½ 
£ CK^{N}N! z^{N}
when zÎ S and N³0.

This function f is a resummation of x^{~}, and it exists if
the opening angle of S is smaller than p.
The formal Borel transform of x^{~}(z) is defined by
y(z)=å_{1}^{¥}x_{n}z^{n1}/(n1)!.
It converges for z<1/c. We assume that the function y
can be continued analytically along a line that does not meet a
singularity. In the particular case when x is a solution of a
linear differential equation with rationnal coefficients, so does y,
as this property is stable under the Borel transform. Thus y has a
finite number of singularities and verifies the above hypothesis. Up
to a possible linear change of variable, we may assume that there is
no singularity on the real axis, which implies that y can be
continued analytically on the positive real axis. If y satisfy the
expected growth conditions at infinity, we apply the Laplace
transform. This transform is defined by
x(z)=L 
(y)= 
ó
õ 

e^{zt}y(t) dt, 
and is convergent for Â(z^{a}) greater than a certain positive
constant, the constant a being made precise later. The asymptotic
expansion of x(z) when z®0^{+} is equal to
å_{1}^{¥}x_{n}z^{n}. The function x is a solution of the
initial differential equation [2, 8].
2 Balser, Lutz, and Schäfke's Technique
The next step is to find a way to compute this function x quickly
and in a large domain. For this, Lutz et al. [1]
reformulate x as a convergent series of the type
x(z)=å_{0}^{¥}d_{n}q_{n}(z). This series is obtained by
introducing a mapping function f that maps [ 0,1 ] onto
[ 0,¥ ], so as to write the equation

x(z)= 
ó
õ 

e^{zt}y°f°f^{1}(t) dt
= 
ó
õ 

e^{zt} 

d_{n}f^{1}(t)^{n} dt,
(1) 
where for the second equality we have used the reexpansion
y°f(u)=å_{0}^{¥}d_{n}u^{n} in terms of the sequence d_{n}.
The sequence q_{n} is thus determined by
q_{n}=ò_{0}^{¥}e^{zt}f^{1}(t)^{n} dt, under the assumption
that the interversion of the integral and the sum holds, permitting
termwise integration. We observe that q_{n} does not depend on x
and on the initial problem, but only on the mapping function
f. This means that these coefficients can be precomputed. On the
other hand the coefficients d_{n} correspond to a composition of the
function f with the Borel transform y. This is formalized in
the following theorem.
Theorem 1 [Balser, Lutz and Schäfke]
Let x(z)=ò_{0}^{¥}e^{(zt)}y(t) dt where the function y
is holomorphic in the domain
DÉ 
{ 
 
Arg(1+t/a) 
 
<p/2p 
} 
and satisfies y(t)e^{bt}®0 as
t®¥ in D. Choose f holomorphic in
D={t<1} so that f(D)ÌD,
f([ 0,1 ])=[ 0,¥ ], and
(1t)^{c}f(t)® A as t® 1 in D. Define
(d_{n}) by its generating series
y(f(t))=å_{0}^{¥}d_{n}t^{n}, and (q_{n}) by
q_{n}(z)= 
ó
õ 

e^{zf(t)}t^{n}f'(t) dt
for z such that Arg(z)<p(1+c)/2. 
Then for suitable positive constants (independent of n)
d_{n}£ Ke^{Lnc/(c+1)} and 
q_{n}(z)£ 

e 
An^{c/(c+1)}Â 
( 
z^{1/(c+1)} 
) 



.

So we have
x(z)=å_{0}^{¥}d_{n}q_{n}(z)
for Â(z^{1/(c+1)}) large.
Proof.
Starting from Equation (1), we obtain
x(z)=ò_{0}^{¥}å_{n=0}^{¥}e^{zt}d_{n}f^{1}(t)^{n} dt.
The saddlepoint method gives upper bounds for d_{n} and q_{n} that
allows us to interchange the order of integrand and summation in the
equation above for Â(z^{1/(c+1)}) large enough. This interchange
yields the expected result x(z)=å_{n=0}^{¥}d_{n}q_{n}(z).
Some other classical conformal mappings can be found
in [6]. Here is an example. The mapping
t=1 

with aÎR
and p³1/2
(2) 
takes the sectorial domain defined by
Arg(1+t/a)<p/(2p) onto the unit disk. The choice of
the conformal mapping f is important because it has an effect on
the speed of convergence and on the area of convergence.
In the particular case where the differential equation is linear with
polynomial coefficients, some efficient computation can be done using
recurrences. We also suppose now that the function f is
algebraic. The precomputation of the coefficients q_{n} is based on
the fact that they follow a linear recurrence. We first note that the
coefficients q_{n} are equal to
ò_{0}^{1}e^{zf(u)}u^{n}f'(u) du as shown by a simple change of
variable t=f(u). The function e^{zf(u)}f'(u)
satisfies the firstorder linear differential equation
G'(t)= 
æ
ç
ç
è 

zf'(t) 
ö
÷
÷
ø 
G(t).
(3) 
If we note å_{k=0}^{K}p_{k}(n)a(n+k)=0 the linear recurrence
satisfied by the Taylor coefficients at the origin a(n) of a power
series solution of the equation (3), then the integrals
q_{n}(z) satisfy the recurrence å_{k=0}^{K}p_{k}(n)q_{nk1}(z)=0.
Once we have the recurrence satisfied by the coefficients q_{n} and
the initial conditions that are given by
q_{n}=ò_{0}^{¥}e^{zt}f^{1}(t)^{n} dt, all the q_{n} can be
computed quickly. A problem is that we seek for numerical and not
exact computations, and so we have, on each example, to seek for
numerical stability. This point uses a backward scheme which is
developped on an example below.
The computation of the coefficients d_{n} can be done efficiently by
finding a recurrence for example using the gfun
package [9], because it is a composition of a known
algebraic function f and a function y known by its differential
equation. The initial conditions for the d_{n} derive directly from
the initial conditions of the differential equation satisfied by y
and so from the initial conditions of the differential equation
satisfied by x^{~}. This is illustrated by the example of the
Heun equation.
3 Heun Equation
The Heun equation is the generic differential equation with four
regular singular points located at 0, 1, c, and ¥;
see [5]. The double confluent Heun equation is obtained
by letting the singularity located at c tend to the one located at
¥, and the singularity located at 1 tend to 0. The
equation obtained then has two irregular singular points located at
0 and ¥. The example we study [3] is the
confluent Heun equation in the form
z^{2}f''(z)+(z+a z^{2}+a)f'(z)+ 
(2a z^{2} b_{1}+
a z^{2}+a^{2} z2g z+2ab_{1}a) 

2z 

f(z)=0.
(4) 
The acceleration is realised by the function f=1/(1z)^{2}1 which
maps from [ 0,1 ] onto [ 0,¥ ].
The recurrence satisfied by q_{n} is thus
q(n)= 
(6+3n)q(n1)+(2z+63n)q(n2)+(n2)q(n3) 

n2 

.
(5) 
The initial conditions, that are easily computed, using the definition
of q_{n}, correspond to a dominated solution, so any numerical error
makes the dominating solution appear. A solution to this problem is to
compute the recurrence backwards, which exchanges the roles of
dominating and dominated regimes. The idea is to choose arbitrary
values for q_{Nd}, ..., q_{N} where d is the order of the
recurrence and N is a sufficiently large integer. All the values of
q_{n} for n£ N are then computed from these ``final'' values
backwards. This technique is developped in [11]. The
dominating solution of Recurrence 5 disappears and
so the initial values found differ only by a multiplicative constant
l from the actual initial values. The sequence q_{n} thus
found has to be multiplied by this constant l to give the
expected sequence q_{n}.
For the coefficients d_{n}, the recurrence is found easily using
gfun. For parameters a=1, b_{1}=1/2,
b_{1}=1/2, and g=1/3, it is
0=(6n^{2}+3n^{3})a_{n}+(93n3675n^{2}18n^{3})a_{n+1}
+(568n+404+267n^{2}+42n^{3})a_{n+2}
+(1193n1176411n^{2}48n^{3})a_{n+3}
+(1042n+1240+291n^{2}+27n^{3})a_{n+4}+(78n^{2}336n4806n^{3})a_{n+5}
with initial conditions a_{0}=0, a_{1}=1, a_{2}=1/3,
a_{3}=23/108, and a_{4}=2749/3888.
Now for each fixed z, we can compute the value of x(z) to
arbitrary
precision,
by choosing the number of terms we take into account. The
backwards computation for the
q_{n} coefficients implies that the number of computable terms is limited
by the starting
point. If it is too low, we have to choose a larger starting point to get
more terms. It is generally not possible to decide where a good
starting point for the computation of the backward computation would
be. This can be
done on particular examples, but the starting point strongly depends on z.
4 Applications
Many problems related to differential equations yield formal power
series of Gevrey order one. Whenever the BorelLaplace transform
applies, the results of Section 2 also applies. A concrete application
coming from physics is the onedimensional complex heat equation:
u_{t}(t,z)=u_{zz}(t,z), u(0,z)=f(z).
The Cauchy data f(z) is assumed to be holomorphic near the origin. A
formal solution is
Lutz et al. have shown that either
u^{~}(t,z) is
convergent, or the method of Section 2
applies. If v(t,z) is the Borel
transform of u^{~}(t,z) with respect to t, then applying
the Laplace
transform in the variable t to v(t,z) for fixed z gives a
convergent solution u(t,z) of the Cauchy problem. Better knowledge
on the function f may easily lead to fast rate convergence possibly
using the mapping function (2).
Another application is about convergent LiouvilleGreen expansions for
second order linear differential equations [4].
References
 [1]

Balser (W.), Lutz (D. A.), and Schäfke (R.). 
On the convergence of Borel approximants. 
Preprint.
 [2]

Borel (Émile). 
Leçons sur les séries divergentes. In Collection de
monographies sur la théorie des fonctions, publiée sous la direction
de M. Émile Borel. 
GauthiersVillars, Paris, 1901. Second edition, 1928.
Reprinted by J. Gabay, 1988.
 [3]

Chyzak (Frédéric), Durand (Marianne), and Salvy (Bruno). 
Borel resummation of divergent series using gfun. 
Maple worksheet. Available from
http://algo.inria.fr/libraries/autocomb/
.
 [4]

Dunster (T. M.), Lutz (D. A.), and Schäfke (R.). 
Convergent LiouvilleGreen expansions for secondorder linear
differential equations, with an application to Bessel functions. Proceedings of the Royal Society. London. Series A, vol. 440, n°1908,
1993, pp. 3754.
 [5]

Duval (Anne) and LodayRichaud (Michèle). 
Kovacic's algorithm and its application to some families of
special functions. Applicable Algebra in Engineering, Communication and
Computing, vol. 3, n°3, 1992, pp. 211246.
 [6]

Kober (H.). 
Dictionary of conformal representations. 
Dover Publications, New York, N. Y., 1952, xvi+208p.
 [7]

LodayRichaud (Michèle). 
Solutions formelles des systèmes différentiels linéaires
méromorphes et sommation. Expositiones Mathematicae, vol. 13,
n°23, 1995, pp. 116162.
 [8]

Ramis (JeanPierre). 
Séries divergentes et théories asymptotiques. Bulletin de la
Société Mathématique de France (Panoramas et Synthèses), vol. 121,
1993, p. 74.
 [9]

Salvy (Bruno) and Zimmermann (Paul). 
Gfun: a Maple package for the manipulation of generating and
holonomic functions in one variable. ACM Transactions on Mathematical
Software, vol. 20, n°2, 1994, pp. 163177.
 [10]

Thomann (Jean). 
Resommation des series formelles. Solutions d'équations
différentielles linéaires ordinaires du second ordre dans le champ
complexe au voisinage de singularités irrégulières. Numerische
Mathematik, vol. 58, n°5, 1990, pp. 503535.
 [11]

Wimp (Jet). 
Computation with recurrence relations. 
Pitman (Advanced Publishing Program), Boston, MA,
1984, xii+310p.
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