Theorem XI-1-12. Suppose that {fn(x)In = 1,2,3,...) is a sequence of contin- uous functions such that they admit unifonn asymptotic expansions
00
a)' (n = 1, 2, 3, ...) as x --+ a in D.
(XI.1.20)
fn(x) ^-- >
M=O
Assume that {f(x)In = 1,2,3.... } converges uniformly to a function f (x) in a subsector Do of D with vertex x = a. Then,
lira Cnm = Cm
00
f (x) ^- E cm(x - a)'
as x -+ a in Do.
M=0
Proof
The assumption implies that for each non-negative integer N, there exists a positive constant KN, independent of n, such that
N
(n = 1,2,3,... )
(XI.1.23)
I
a ) ' 1: 5
M=0
for x in a neighborhood of a in D. Furthermore, for each pair of positive integers (j, k), there exists a positive constant bjk such that
(XI.1.24)
If,(x) - fk(x)I S bjk
for x in Do, where
(XI.1.25)
bjk-+0
as j, k -+ oo.
Put
N
(XI.1.26)
pjN(x) =E cjm(x - a)n'
(j = 1, 2, 3, ... ).
M=0
1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCARE 349
Then,
(XI.1.27)
Ifj(x) -P)N(x)I 5 KNIx-alN+1
(j = 1,2,3,...).
Thus,
(XI.1.28)
IP) N(x)-PkN(x)I 5 5)k+2KNIx-alN+l
(j,k=1,2,3,...).
In particular,
(XI.1.29)
Ic)o - ckol < b)k+2Kolx-al
(j,k= 1,2,3,...)
for N = 0 and x in Do. Therefore, (XI.1.25) and (XI.1.29) imply that lim c)o = CO
exists.
Assume that lim c),,, = c,,, exist for m < N. Then, from (XI.1.28), it follows
1 00
that
N-1
E (C),,, - Ckm)(x - a)m + (CjN - CkN)(x - a)N
M=0
< S)k + 2KNIx - alN+1
(j,k = 1,2,3,...)
for x in Do. Thus,
IC)N - CkNI Ix - aIN < e)k + 61k + 2KNIx - alN+1
k=1 ,2 , 3 . . . .
where e)k 0 as j, k -. oo. Hence,
Ic)N -CkNl :5- 11 aIN +2KNIx-al
(j,k= 1,2,3,...).
Iz
Therefore, lim c)N = cN. Consequently, lim c),, = c,,, for all m.
) --00 l-00
Furthermore, since (XI.1.27) holds independently of j, we obtain
N
f(x) - > cm(x - a)m < KNlx - alN+1
M=0
for x in Do. Thus, (XI.1.22) holds.
The following basic theorem is due to E. Borel and J. F. Ritt.
Theorem XI-1-13 ((Bor( and [Ril). For a given formal power series in (x - a)
(XI.1.30)
P(x) = > c,,,(x - a)m
m=0
and a sector with vertex x = a
(XI.1.31)
D={xIO<Ix -al <ry, (31<arg(x-a)<p2},
350 XI. ASYMPTOTIC EXPANSIONS
them exists a function f (x) which is continuous in D and holomorphic in the inte- rior of D, and
00
(XI.1.32)
f (x) ^_- E c,,,(x - a)' as x - a in D.
M=0
Proof.
Without loss of generality, assume that the sector D contains the ray arg(x - a) = 0. Construct functions a,,, (x) for m = 1, 2.... such that f (x) = co +
00
E ca,,,(x)(x - a)m is continuous in D, holomorphic in the interior of D, and
m=1
satisfies (XI.1.32). Let bm and 0 be positive numbers, where bm are to be specified later, whereas 0 is chosen to satisfy 0 < 0 < 1 and
(XI.1.33)
01 arg(x - a)I < 2
for x E D.
Consider
r
1
(XI.1.34)
am(x) = 1 - exp l - 2m1,m(x - a)0 J , m = 1, 2, ... .
Then, am(x) are holomorphic in a `sector containing D. Also, from (XI.1.33) and
the fact that
I1-esl=l jetdt <Izi
for
z<0,
it follows that
(XI . 1 . 35)
bm
m= 1 , 2 ,...,
lam(x)I <
- aIe'
- 2mymlx
in D. Put
(XI.1.36)
bm = I Icml-'
if c,,, 0 0,
0
if Cm=0.
Then,
00
00
Ix - alm-e
Icmam(x) (x - a)mI < E
for x E D.
M=1
m=1
2m.ym
Hence,
Oo
f (x) = co + E c,nam(x) (x - a)m m=1
converges uniformly in D. Consequently, f (x) is continuous in D and holomorphic in the interior of D.
1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCARE 351
To show that f (x) satisfies (XI.1.32), let N be a positive integer and observe that
N+1
N+1
AX) -0D - E cm(x - a)m = E C exp
2--y-(Z -
ax - a)m
m=1
m=1
7)01(
+ (x - a)N+1
00
i cmam(x)(x -a) m-N-1] .
Lm=N+2
Since (XI.1.33) implies that
iexp [
] Ix - aj-"I,
r = 0. 1, 2,... , N,
2'n7m(x - a)e
are bounded for x E D, there exists a positive constant H1 such that
N
r _
bm
<Hljx-aIN+1 for xED.
,,=1
c-exp I
2mrym(x - a)e] (x - a)m
m-1
Also, (XI.I.35) implies that there exists a positive constant H2 such that
00
1
Ix
-aim-N-1
E cmam(x) (x -
a)m-N-1
[00
(27)N+1Ix - aj0
m=N+2
(2'r)m-N-1
m=N+2
N+2
I {x - aI -
H2
for X E D.
- i (2y)
+2
2y
Thus, there exists a positive constant KN such that
f(x)-
N
c,n(x-a)m<KNIx-aIN+1
for xED.
Therefore, (XI.1.32) is satisfied. 0
Similarly to Theorem XI-1-13, we can prove the following theorem.
Theorem XI-1-14. For a given formal series p(x, t) _ 00 cm(t)(x - a)m in pour
m
ers of (x - a and a sector D given by (XI.1.31), where q. (t) (m = 0, 1, ...) are
holomorphic and are bounded in a domain S2, there exists a function f (z, t) which is holomorphic with respect to (x, t) in D x S2 and satisfies the conditions
(XI.1.37) A x , t) c,,,(t)(x - a)m as x -+ a in D
M=0
352
XI. ASYMPTOTIC EXPANSIONS
and
( X ,
00
as x -. a in
V
(XI.1.38)
dt
m=o
dt
uniformly fort in Il.
Proof.
As
I dcd't(t) are bounded in Sl (m = 0,1, ... ), choose
{
I ax 1cm(t)I] + I
if c -(t)
0,
6m =
t r: I d d (t) J }-1
0
L
`
ifC,,,(t)-0.
Set
00
f (x, t) = co(t) +
Cm(t)am(x)(x - a)m,
m=1
where a,,, (x) are given in (XI.1.34). Then, it can be shown that f (x, t) is holomorphic with respect to (x, t) in D x 1 and satisfies (XI.1.37) and (XI.1.38) in a manner similar to the proof of Theorem XI-1-13. O
Examples XI-1-15. The followings are two examples of functions admitting as- ymptotic expansions.
1. Let r(z) denote the Gamma function and Log z denote the principal branch of the complex natural logarithm of z. If 0 = arg z satisfies IOI < 7r, a real number w
satisfies Iw I < 2 , and 10 + w l < 2 , then
Log[r(z)] _ (z - Logz - x +
rN m-1
M=1 2m(2m
BN+1
-(2N+1)
+ eN(z) 2(N+1)(2N+1)z
where IeN(z) < (cos(8 + w))2N+1I sin(2w)I
and the B,,, are the Bernoulli numbers
(see, for example, [01, p. 294]).
2. The exponential integral function Ei(z) = J zo0 ettdt has the following form:
N
Z
Ei(z) = er J(k - 1)!z-k + N!100 ett-N-ldt.
k=1
From this observation, it follows that
N
e-ZEi(x) - E (k -
1)!z-kI
< {N! + (N + 1)!(7r + (N + 1)-1)) IzI-N-1
k=1
for I arg(-z)I < 7r. The asymptotic expansion is valid for I arg(-z)I < 32 (see, for
example, (Wasl, p. 31[).
2. GEVREY ASYMPTOTICS
353
XI-2. Gevrey asymptotics
The Gevrey asymptotics is based on the following two definitions.
Definition XI-2-1. Let s be a non-negative number. A formal power series p =
+oo
E anx' E C[[x]) is said to be of Gevrey order s if there exist two non-negative
m=0
numbers C and A such that
(XI.2.1)
[am[ < C(m!)'Am
for m = 0,1,2,... .
We denote by C([x]], the set of all power series of Gevrey order s.
Definition XI-2-2. A function 0 of x is said to admit an asymptotic expansion of Gevrey orders (s > 0) as x -+ 0 on a sector
D(r, a, b) = {x : 0 < fix[ < r, a < arg(x) < b},
where a and b are two real numbers such that a < b and r is a positive number if
(i) ¢ is holomorphic on V (r, a, b),
+00
(ii) there exists a formal power series p = > a,,,x"' E G[[x]], such that an in-
equality
m
N-1
(XI.2.2)
E amxm < Kv,a,A(N!)'(Bv,a,3)8[x[N
0(x) - M=0
holds on D(p, a, 0) for every positive integer N and every (p, a, 03) satisfying the inequalities 0 < p < r and a < a <,3 < b, when KP,Q,o and Bp,,,,o are non-negative numbers determined by (p, a, /3).
We denote by A, (r, a, b) the set of all functions admitting asymptotic expansions of Gevrey order s as x -' 0 on the sector D(r, a, b). We also set J(4) = p for
0 E A. (r, a, b). In §XI-3, we shall explain the basic properties of ar[[x]] A. (r, a, b), and the map J : A. (r, a, b) -+ G[[x]],.
In the example given at the beginning of this chapter, the formal solution p =
00
E (-1)m(m!)x"+1 of X2!Ly +y-x= 0 is of Gevrey order 1 and the solution f (x) =
M=0
eL/z Jxt-1e-lltdt admits an asymptotic expansion of Gevrey order 1. Furthermore,
JJa
J(f) = p. Also, the Maillet Theorem (cf. Theorem V-1-5) states that any formal solutions of an algebraic differential equation belong to C[[x]], for some s, where s depends on each solution.
It is well known that if a complex-valued function 0(x) is holomorphic and bounded on a domain 0 < [xj < r, where r is a positive number, then 0 is repre-
sented by a convergent power series in x. The Gevrey asymptotic expansions arise
354
XI. ASYMPTOTICS EXPANSIONS
in a similar but more general situation. To explain such a situation, let us consider N sectors
St={x: 0<Ixl <r, at<arg(x)<b1}
(1=1,2,...,N)
N
which satisfy the condition USt = (x: 0 < IxI < r}. The set {S11S2,... SN} is
t=1
called a covering at x = 0. Also, a covering {S1, S2, ... , SN } at x = 0 is said to be good if
(i) at < at+l (t = 1, 2, ... , N), where aN+l = al + 27r,
(ii) bt - at < x (1 = 1, 2,... , N),
( ) S t n St+1 0 (1 = 1, 2, ... , N) and Stn Sk = 0 otherwise if 1,,E k, where
SN+1 = S1.
The following theorem is the most basic fact in the Gevrey asymptotics.
Theorem XI-2-3. Assume that a covering {Si, S2, ... , S1 %r) at x = 0 is good and that N functions 01(x), ¢2 (x), ... -ON (x) satisfy the conditions
(1)4t(x) is holomorphic on St,
(2)dt(z) is bounded on St,
(3)it holds that
(XI.2.3)
I¢e(x) - 01+1(x)I < 'Yexp [-
on St n St+1,
where -y > 0, A > 0, and k > 0 are suitable numbers independent of 1.
Set
(X1.2.4)
+00
Then, there ex,sts a formal power series p = E a ..xm E C[[x)]. such that ¢t E
M=0
A. (r, at, bt) and J(41) = p for each 1.
There are various situations in which Gevrey asymptotic expansions arise. To illustrate such a situation, let ¢(x) be a convergent power series in x with coefficients in C. For two positive numbers r and k, set
krr
xJ0 0(t)e-('1x)}tk- 1 dt.
This integral is called an incomplete Leroy transform ofd of order k.
+oo
Theorem XI-2-4. For every ¢ =
cx"j E C{x}, it holds that
m=0
R if
4,k(0) E Al/k P,-2k'2k
2. GEVREY ASYMPTOTICS
355
and
+00
J(tr,k(4)) _
r 1 1 + k Cm2'",
M=0
where k and p are any positive numbers and r is any positive number smaller than the radius of convergence of 0.
Proof.
The following proof is suggested by B. L. J. Braaksma. For an arbitrary small positive number e, let {Sl (e), S2(e), ... , SN(c)} be a good covering at x = 0, where
St(E) _ {x : I argx - dtI < 2k - e,
0 < jxj < r}
with real numbers dt such that -ir < d 1 < . - - < dN
ir, and set
J OtW _
(t = 1, 2, ... , N).
I. 4t(x) = tr,k(0t)(xe-d`)
In particular, choose dt = 0 for some t. Then, {Sj(e),S2(e),... ,SN(e)) and
{01, 02, . - . , ON} satisfy conditions (1), (2), and (3) of Theorem XI-2-3. In fact, (1) and (2) are evident. To prove (3), note that
k
/ re'd'
r
Qe(x) =
J
0(a)e-(o/s) o 'da,
zk
0
where the path of integration is the line segment connecting 0 to re{d'. Therefore,
of (x) - bt+1(x) =
k
jt
0(a)e-(ol=)'ak-1d,
X
where the path ryt of integration is the circular arc connecting re'd'+1 to re'd'.
Statement (3) follows, since
e-(°/=)`exp - t/fx cos[k(arg a - arg x)J
and kj arga-argxj < 2 -ke for a E ?Y and x E St(e)f1St+1(E) Since a is arbitrary, it follows that
61 E Allk I P, de - r,-, dt +
(t = 1, 2, ... , N).
2k
Furthermore, J(01) can be computed easily (cf. Exercise XI-13).
+00
Observe that a power series p = 1: a,nxm E C[[xjj belongs to C[[xJ), if and only
m=0
+ao
if d(x) =
I'(l sm)xm belongs to C{x}. Therefore, we obtain an important
corollary of
Theorem XI-2-4.
+
356
XI. ASYMPTOTICS EXPANSIONS
Corollary XI-2-5. For any p E G[[x]], and any real number d, there exists a function ¢(x) E A. (r, d - 2 , d + ) such that J(O) = p.
This corollary corresponds to the2Borel-Ritt Theorem (cf. Theorem XI-1-13) of the Poincar6 asymptotics. Also, this corollary implies that the map J
A8 (r, d - s7r2 , d + 2) -+ c[[x]], is onto.
Theorem XI-2-3 is a corollary of the following lemma.
Lemma XI-2-6. Assume that a covering {St : e = 1, 2,... , N} at x = 0 is good and that N functions 61(x), 62(x), ... , 6v (x) satisfy the following conditions:
(i) 6t is holomorphic on St n St+1,
(ii) I6t(x)I 7exp[-Alxl-k] on St n St+1, where -y > 0, A > 0, and k > 0 are suitable numbers independent of e.
Define s by (XI. 2-4). Then, there exist N functions ), (x), t.b2(x), ... ,'IPN(x) and a
+oo
formal power series p = >
E C[[x]], such that
m=o
(a)4,t E A. (r, at, bt) and J(4't) = p, where St = {x : 0 < I x[ < r, at < arg(x) < bt} (e = 1, 2, ... , N),
(b) 6t(x) _ t(x) - i/'e+l(x) onSt n St+1.
Let us prove Theorem XI-2-3 by using Lemma XI-2-6.
Proof.
Set
5t(x) = 4t(x) - 41+1(x)
(e = 1, 2, ... , N).
Then, there exist N functions 4111(x), 02(z), ... , ON (x) satisfying conditions (a) and
(b) of Lemma XI-2-6. In particular, (b) implies that
Oe(x) - 4,t+1(x) = t t(x) - 4Gt+1(x)
(e = 1, 2,... , N)
on St n St+1. This, in turn, implies that
01(x) - 4't(x) = 4,1+1(x) - 4Gt+1(x)
(e = 1, 2,... , N)
on St n St+1. Define a function 0 by
4,(x) = 4t(x) - 4Vt(x)
on St (e = 1, 2, ... , N).
Then, 0 is holomorphic and bounded for 0 < IxI < r. Therefore, 0 is represented
by a convergent power series. Since .01 = -01 + 0, Theorem XI-2-3 follows immedi-
ately. 0
We shall prove Lemma XI-2-6 in §XI-5.
Because the Gevrey asymptotics of functions containing parameters will be used later, we state the following two definitions.
3. FLAT FUNCTIONS IN THE GEVREY ASYMPTOTICS
357
00
Definition XI-2-7. Lets be a positive number. A formal power series
am(u-')em
m=0
is said to be of Gevrey order s uniformly on a domain V in the u-space if there exist two non-negative numbers Co and C, such that
(XI.2.5)
Iam(i )I < Co(m!)sCr
for u E D and rn = 0,
1, 2, ....
Set V = D(60, a0, Qo) = {e : 0 < Ie] < 5o, ao < arg e < (30} and W = D(b, a, Q).
Definition XI-2-8. Let s be a positive number. A function f (u e) is said to admit
00
an asymptotic expansion
am(u")em of Gevrey order s as c
0 in V uniformly
m=0
on D if
(i) > am(i )em is of Gevrey order s uniformly on D,
m=0
(ii)for each W such that a0 < a < 0 < X30 and 0 < b < 60, there exist two non-negative numbers KW and Lµ such that
N
(X1.2.6)
f (u, e) - > am('tL)cm <
1)!]8L
M=0
fori ED,eEIV andN=1, 2, ....
Theorem XI-2-3, Theorem XI-2-4, Corollary XI-2-5, and Lemma XI-2-6 can be extended in a natural way so that we can use them for functions containing parameters. We leave such details to the reader as an exercise.
The materials of this section are also found in [Rain 11, [Ram 21, [Si17, Appen- dices], and [Si18).
XI-3. Flat functions in the Gevrey asymptotics
In the next section, we shall show that C[[x)), and A. (r, a, b) are differential
algebras over C and the map J : A, (r, a, b)
C[[x)), is a homomorphism of
differential algebras over C. In §XI-2, it was shown that the map J is onto if b - a < sir (cf. Corollary XI-2-5). In this section, we explain the basic results concerning the nullspace of J. To begin with, we introduce the following definition.
Definition XI-3-1. A function f (ii, e) is said to be flat of Gevrey order s as e - 0
in a sector
V = D(r,a,b)={e: 0<je]<r, a<arg(e)<b)
uniformly on a domain D in the u'-space if f (ii, e) admits an asymptotic expansion
+00
p = E am (u")em of Gevrey orders as f 0 in V uniformly on D and the expansion
m=0
off is 0, i.e., all the coefficients a, (ii) of p are equal to zero.
The following theorem characterizes flat functions of Gevrey order s.
