Hua C., Wong R. Differential Equations and Asymptotic Theory in Mathematical Physics
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GEVREY ASYMPTOTICS AND APPLICATIONS TO
HOLOMORPHIC ORDINARY DIFFERENTIAL EQUATIONS
JEAN-PIERRE RAMIS
Lab.
E.
PICARD (UMR
-
CNRS
5580)
and Inst. Univ. de France
UFR
MIG,
Universite' Paul Sabatier de Toulouse
118,
route
de
Narbonne
31062
Toulouse
cedex
4,
France
E-mail: ramis@picard.ups-tlse.
fr
0.
Introduction
Our
main
problem
is the quantitative and qualitative study of solutions
of
functional equations.
We will mainly consider the case of
one
indepen-
dent variable, the unknown being a vector variable.
Historically it begins with
algebraic
functional equations. Here are some
examples:
1
y"
+
y
=
0,
y'
=
-,
Zf(Z)
=
f(x
f
l),
f(2Z)
=
f(Z)2,
. . .
X
(homogeneous linear ordinary differential equations, non-homogeneous lin-
ear ordinary differential equations, homogeneous linear difference equations,
q-difference equations,
. .
.).
Later mathematicians studied
analytic
func-
tional equations like
Z"
+
(sint)x
=
0
(pendulum equation), and later
C", C",
. . .
equations.
In these notes, we will limit ourselves to the case
of
analytic
equations.
Our technics are efficient
only
in this case, which is already very difficult
and very interesting for the
applications.
Our main concern will be a
local study,
the study of solutions of
analytic functional equations in a neighborhood of a point. This point will
be a
regular
point or
a
singular
point, the second case being the more
44
45
difficult but the more interesting. It is extremely important to distinguish
between
formal
(or
symbolic) solutions
and
actual solutions.
Formal Solutions.
We search (by hand or using
a
computer algebra
system like
Maple)
symbolic solutions, like
formal power series
solutions
n=O
or more general solutions like
Cm
n=O
Typically a pair (Functional Equation
+
Initial Condition) gives rise to a
program
generating a sequence of coefficients
(a0
,
a1
,
.
. .
,
a,,
. . .
)
.
Actual Solutions.
There are different ways to handle actual solutions.
We can use a
theoretical approach
(using in general
Functional Analysis)
or
a numerical and graphical approach (with
a
computer), typically a Runge-
Kutta method for Ordinary Differential Equations.
Our aim is to build locally (in a neighborhood of a regular or singular
point) a
good dictionary
between
Formal
and
Actual
Solutions:
Formal solutions
++
Actual sdutions
The elementary situation.
In each case, we will first consider “reg-
ular points”. The
formal
power series solutions
at
a
regular point will
converge automatically,
and, if
f
is the
classical sum
of the formal power
series solution
f,
then
f
will be an
actual solution.
Example
0.1.
Cauchy theorem for Ordinary Differential Equation (equa-
tions or systems)
_-
dY
-
F(~,Y)
(F
analytic)
dx
A nearly elementary situation.
Some singular parts of analytic
functional equations are “not too bad”. In some sense they are similar
to
regular points: more complicated symbolically but not analytically.
At
these points the power series expansions appearing in symbolic solutions
will automatically
converge.
A
very important example of this situation and
historically the first one is the
Fuchs theory
for linear ordinary differential
46
equation discovered by the German mathematician
L.
Fuchs in XIX-th
centaury.
A
linear analytic equation
fuchsian
(or
regular singular)
at
the
origin of the
complex plane
can be written:
d
d
dx dx
(z-yy
+
un-l(z)(z-)n-ly
+
.
.
.
+
uo(z)y
=
0,
the
ao,
a1
, . . .
,
an-l
being holomorphic near zero.
ferential Equation of Euler and Gauss:
An extremely important example is the hypergeometric Ordinary Dif-
E(u,
b,
c,
2)
z(1
-
z)ytt
+
[c
-
(u
+
b
+
l)z]y’
-
aby
=
0.
It
admits singularities
on
the Riemann sphere
P1(C)
=
CU{co}
only at
the points
0,
1,
00,
and these singularities are
fuchsian.
The difficult situation.
When we have eliminated regular points (the
“generic” case) and regular-singular points, it remains (perhaps) the
so-
called
irregular singular
points.
At
such points the power series expansions
appearing in the symbolic solutions are “in general”
divergent
(they will
converge ‘Lexceptionally’’).
Example
0.2.
For the following equation
x2yt
+
y
=
0
we have a symbolic (and actual
-
. .
)
solution
el/”,
its coefficient is the power
series
f
=
1
which of course converges
!
Example
0.3.
The equation
x2yl+
y
=
z
will be called
Euler Equation.
It
was introduced by
L.
Euler during the
+m
XVIII-th century when he tried to give a sense to the expansion
C
(-l)%!
n=O
(and reached some success in this attempt
.-.).
Euler differential equation admits a
unique
power series expansion
so-
lution
(at
zero): the Euler series
4-m
This series is evidently
Equation is of course
n=O
divergent.
The general symbolic solution of Euler
f(z)
+
cel/“ (c
E
C).
47
There are many such examples in special functions theory: cf. the sym-
bolic solutions of Borel, Kummer, Whittaker ordinary differential equations
at infinity
[13], [34].
This divergence phenomena is puzzling.
It
is of course a serious problem,
but it is very important because it appears in applications, in particular
in physics (caustics, energy levels of hydrogen atom
.
. .
).
Finally as we
will see it is possible to overcome completely the difficulty. Moreover there
is a very interesting phenomena: in some case, when one performs
actual
computations
with
divergent
series, one gets
a
lot more precise results (nu-
merically) than with
convergent
series. This was discovered by Stokes' for
the computation
of
fringes for optical caustics [24]. The reader will find an
extremely striking example in
[8].
1.
Asymptotic
expansions
1.1.
Classical asymptotics
(or
Poincare' asymptotics)
Asymptotics were introduced by the French mathematician
H.
Poincard.
It
is worth to go back to his original paper
[22]
(1882). Here we will start
from the introduction of
Me'canique Ce'leste,
part
2
[23].
It
is important to
notice that the notion of
classical asymptotics
or'
Poincare' asymptotics
in
some sense misses the original problems risen by PoincarB.
We start from a power series expansion
formal
x:2un~n
E
c[[z~],
purely (the
an's
are
arbitrary complex
numbers). The problem is to give a
reasonable sense to the sum of the series:
+m
C
u,xn
=
Some
actual function
This problem rises from the fact that, in some cases, formal power series
solutions are solutions of ordinary differential equation or partial differential
equation, but are divergent (cf. Section
0).
This happens in particular in
applications, like Celestial mechanics (astronomy). In such cases, we can
hope that the summation process will give an actual solution
of
the ordinary
differential equation or the partial differential equation. We will see that it
is possible to get such processes, but only after a hard work.
n=O
In [23],
H.
Poincar6 describes
two
summation processes:
lcf.
below
1.2
48
1.
Summation “in the geometers sense”: it is the
usual sum
of
conver-
+m
gent
power series expansions
C
anxn
E
C{x},
when
lanl
<
CA”
for some
n=O
positive constants
C,
A
>
0,
independently of
n
E
N
(Geometric growth).
2.
Summation “in the astronomers sense”: it is an
old method,
already
used by Euler, Lacroix
. . .
,
called the
summation at the smallest term.
The principle is the following. The following situation appears in many
+cc
applications. The power series expansion
C
a,xn
is
divergent,
but
lanxnl
decays from
n
=
0
to
n
=
N,
and growth with
n
for
n
2
N.
n=O
(
X
X
X
X
N
Fig 1.1.3
Be careful here:
N
=
N(x)
depends
on
x.
Then we can
decide
that the
N
+m
finite
sum
C
anxn
E
C
is the sum
of the power series expansion
C
anxn.
n=O
n=O
A
typical example discussed by Poincark
[23]
(and related to Euler series,
Im
1
0-
cf
0.
Introduction) is the power series
C
n!xn
with
x
=
-!--
Here
n=O
1000‘
an+
lxn+
anxn
a,
=
n!.
We have
In
=
;
0
<
In
5
1
for
n
+
1
5
1000,
In
2
1
for
1
n
+
1
2
1000. The smallest term corresponds
to
n
=
N
=
1000
=
(For
general values, if
x
>
0,
we will choose the entire part
(-1
of
-).
Using
Stokes description of similar power series, we can say that the series
first
converqes
and
after diverqes.
X
11
xx
+m
xn
n=O
n!
If now we consider the power series
C
-
with
x
=
1000,
we see that
X”
n.
lanxnI
=
7
first increases with
n
=
N
=
1000 and after decreases. This
49
series is convergent in the usual (geometers) sense. In Stokes style we can
say that it is
firstly divergent
and
after convergent.
H.
Poincark stresses upon the fact that the summation process
2
(as-
tronomers) is in many
practical
situations
numerically very
accurate: the
power series expansion
C
a,xn
represents symbolically
an
actual function
+a
n=O
+m
f
and the
sum
C
anxn (at the smallest term) gives a
very precise
nu-
n=O
merical value of f(x).
It
is quite surprising that this method based upon
divergent expansions
and unfounded theoretically (up to now, cf. below) is
more efficient for numerical computations than the use of
convergent
power
series expansion. (cf. the striking example of computation of fringes near a
caustic by Stokes
[24].)
We will explain below why in the classical example
of the integral (cf.
[all)
We return now to Poincark ideas. He was convinced that a formal power
series expansion which is a (formal) solution of
a
problem (from mathemat-
ics, physics, astronomy,
.
.
.
stated in
analytical
terms)
must represent
an
actual function solution
of the problem
(holomorphic
on some convenient
domain). Apparently this idea has been shared by many scientists before
Poincark: Euler, Cauchy, Briot and Bouquet, Stokes
. . .
,
but some impor-
tant mathematicians in Poincar6 times were thinking the contrary (like
E.
Picard). This conviction of Poincart. led him to his fundamental definition
of asymptotics'.
Let
C
unxn
E
@[[x]]
be an arbitrary power series expansion. Let
V
be
n=O
an
open sector
in
@*
=
@
-
{0},
with its vertex at the origin
0
E
@:
V
=
{x
E
@*/IxI
<
R,argx
E
(a,,B)}.
(More generally we will consider sector on the Riemann surface of the log-
arithm, that is why we will allow
I,B
-
a1
>
27r.)
We will denote by
O(V)
the @-algebra of
holomorphic
scalar functions on
V.
2We insist on the fact that with this definition Poincar6
missed completely
the subtle
phenomenon
of
the summation at the smallest term.
50
Fig
1.1.2
Definition
1.1.
Let
f
E
O(V) (holomorphic on the open sector V). Let
f
E
C
anz"
E
@[[z]].
We will say that
f
is asymptotic to
f
on V (in the
classical
or
Poincark's sense) if, for every strict3 subsector
W
4
V, there
exists some positive constants
1Mw,"(n
E
N)
such that, for every
z
E
W,
the following estimates hold:
+m
n=O
I
p=o
I
We will denote
f
E
d(V) and
f
-
f
on V. The set d(V) of functions
in O(V) admitting an asymptotic expansion is clearly a sub-@-algebra
of O(V). Moreover, if
f
E
d(V) admits
f
E
@[[z]]
as an asymptotic
expansion, then its derivative
f'
admits
f^'
as an asymptotic expansion.
For elementary properties of asymptotic expansions, we refer the reader to
W.Wasow's book
[33].
The asymptotic expansion, when it exists is clearly
unique.
The Taylor
map
J
:
d(v)
+
@[[z]],
f
H
f
is an homomorphism of differential
@-
algebras. The basic idea of
H.
Poincark was that if
f
E
d(v)
admits
f
E
@[[XI]
as
an asymptotic expansion, then in some sense, we can consider
f
as
a sum of
f.
Unfortunately such
a
sum
is
not unique. The non-uniqueness
corresponds to the "Error Space" kerJ
=
d"O(V), which
is
the differential
ideal of d(V) of the infinitely flat holomorphic functions on
V.
Example
1.1.
V
=
{Rex
>
0},
f(z)
=
exp-S,
f
N
0.
3A
subsector
W
C
V
is
strict
if
w
-
(0)
c
V
(w
being
the closure
of
W
in
C).
51
We insist on the derivation property.
It
works because we deal with
holomorphic functions and because in the definition of asymptotics we asked
uniqueness only on strict subsectors.
The proof follows immediately from the following lemma.
Lemma
1.1.
Let
g
E
O(V).
IJ
for
every strict subsectors
W
4
V,
we
have estimates
Ig(x)l
<
Cw(xln+’,
then we have similar estimates on the
derivative:
Ig’(x)l
<
CbIxln.
Idea of the
proof.
Let
W’
4
W”
4
V. We denote by
D(x,d)
the
closed disc (in
C)
with center
x
and radius
d
>
0.
There exists
b
>
0
such
that, for every
x
E
W’,
we have
D(x,blxl)
c
W”.
Then we can write a
Cauchy integral
and our lemma follows.
have
It
is easy to prove that
f
E
d(V)
if and only if, for any
W
+
V,
we
+==
Then
f
-
f^
=
C
a,P.
n=O
We have seen that kerJ
=
d<O(V) is an obstruction for the uniqueness
of our “summation”. What about the “existence”, that is the surjectivity
of
J
:
d(V)
-+
@[[XI].
Is it possible to “sum” any power series expansion
f^
E
C[[x]]
using asymp-
totics? The answer is
YES.
Theorem
1.1.
(Bore1
-
Ritt
Theorem) The Taylor map
J
:
d(V)
4
@[[XI]
is surjective. (It extends to any “ramified sector”
V.)
Proof.
cf.
[33]
(Theorem
9.3,
p.
43).
If
.f
E
R[[x]],
we can choose
f
E
d(V) real on the real axis.
We can summarize the situation by the short exact sequence:
0
--f
d<O(V)
4
d(V)
5
C[[x]]
+
0