Theorem XI-3-2. A function f (u", c) is flat of Gevrey order s as e -- 0 in a sector D(r, a, b) uniformly on a domain D in the ii-space if and only if for each
W = D(p, a, Q) such that 0 < p < r and a < c< < ft < b, there exist non-negativenumbers KW and Aw such that
(XI.3.1)
If(u,e)I<KtyexpI-ikI for (i,e)EDxW,
where
(XI.3.2)
I.
k =
s
Proof.
Suppose that for a fixed W, there exist two non-negative numbers C and A such that
I f ( u " , e)j < C(m!)"Am jelm
for
(11,f) E D x W and m=0,1,2,....
Then, by virtue of Stirling's formula
r`k)
2m
(XI-3.3)
m! = 2nm mme'm l[i +0(01,
If(u,e)I < Com"'Ag'jem for
(u, e) E D x W and m = 0, 1, 2, .. .
For a given e E W, choose a non-negative integer m so that
k
me <
(;i)
< (m+1)e.
Then,
1
I f(u' e) 5
Comm/k (me)
= Coe-"'/k = Co exp I ke {e - (m + 1)e}]
Hence, I f (u", e) I < K exp[-Aej-k] for every (6,e) ED x W, where K = Coexp[s] and A = . The converse is evident. 0
e 0
3. FLAT FUNCTIONS IN THE GEVREY ASYMPTOTICS
359
Remark XI-3-3. It is known that a function f (11, e) is identically zero on D x V if
(i)f is holomorphic in a on V for each fixed ii in D,
(ii)f is flat of Gevrey order s as e --+ 0 in V uniformly on D,
This implies that the homomorphism J :A, (r, a, b) - C[[xBJ, is one-to-one if b - a > sa. Remark XI-3-3 is a consequence of the following lemma.
Lemma XI-3-4 ([Wat]). Let V = D A - T,
), when p is a positive number.
2k
2k
If f (x) is holomorphic in V and If (x)1 < Cexp[-Blxl-kJ for x E V for some positive numbers k, B, and C, then f = 0 identically in V.
Proof
Fora>k,set
B
F(x,a) = f(x) P cos(k ) x_k,2a
Then, for each fixed value of a, F is holomorphic in V and
IF(x,a)l < Cexp - B C1 - coss((k9) (k7r)) IxE-k I
for x E V,
where 0 = argx. In particular,
IF(x,a)l < C
for
argx = ±7r,
x E V,
IF(x,a)1 < Cexp IB ( cos()-E- 1 r-k
1
for Ixj = r <pand IargxI <
Therefore, by virtue of the Phragmen-Lindelof Theorem (cf. Lemma XI-3-5), we conclude that
IF(x,a)I < Cexp [B (COS I
for
I arg x1 < , x E V.
cos(U')
In particular,
If (x)I < C exp [BI
i)p-k - COS
) sk]
for argx = 0, x E V.
Letting a -+ k, we derive f (x) = 0 if argx = 0 and x < p. This completes the proof of Lemma 111-3-5.
Since use was made of the Phragmdn-Lindelof Theorem, we shall state and prove the theorem precisely (cf. [Ne2, pp. 43-44J).
360
XI. ASYMPTOTICS EXPANSIONS
Lemma XI-3-5(Phragmen-Lindelof).Let W be a closed sector in C given by
W = {x : a < arg x < ,Q, 0 < IxI < p(arg x) },
where p(w) is a positive-valued and continuous function on the interval a < w < Q. Assume that
(1)Q - a < A , where k is a positive number,
(2)f is continuous on W,
00
(3)f is holomorphic in W, where W is the interior of W,
(4)I f(x)I < Kexp[AIxI-k] for x E W, where A and K are positive numbers,
(5)If (x) I < M for x on the boundary of W except for x = 0 , where M is a
positive number.
Then,
(XI.3.4) If (x)I < M for x E W.
Proof.
Choose I > k so that Q - a < a < 7r. Set g(x) = f (x) exp[-e(e-'sx)-r], where
0= a
P 1.
Q and e> 0. Then,
2
I9(x)I = If(x)Iexp[-ecos(t(argx - 0))Ixl-`]
< Kexp[-IxI-r(ecos(f(argx - 0)) - AIxIt'k)]
(3- a)
for x E W. Note that eI arg x - 01 < f J < 2 for x E W. Therefore,
2
lim g(x) = 0 and I9(x)I 5 If(x)I for x E W. Since g is holomorphic in W, it
Izl-u
follows that Ig(x)I < M for x E W. Therefore,
If (x)I < 11f lexp(e(e-sex)-t)I
for
x E W .
Letting e - 0, we derive (XI.3.4).
We can also prove the following theorem without any complication.
Theorem XI-3-6. If f is holomorphic in e on D(r, a, b) for each fixed u" in V and f is flat of Gevrey orders as e - 0 in D(r, a, b) uniformly on a domain D in the
u"-space, then
j
f (u, a)do and of (a, e)
are also flat of Cevney orders as e -' 0 in
Of
a, b)
ly on D.
The proof of this theorem is left to the reader as an exercise.
For more information see [R.aml], [Ram2] and [Si17, Appendices].
XI-4. Basic properties of Gevrey asymptotic expansions
In this section, we summarize the basic properties of the Gevrey asymptotic
expansions. First we prove the following theorem.
4. BASIC PROPERTIES OF GEVREY ASYMPTOTIC EXPANSIONS 361
Theorem XI-4-1. Assume that
(i)a C-valued function F(ul,... , un, e) is holomorphic in (ul,... , un, e) in a do- main D(rl,... ,rn) x D(r,a,b), where D(rl,... ,rn) = {(u1,... ,un) : Iu3f <
r,, j = 1, ... n},
(ii) flu l, ... , un, e) admits an asymptotic expansion p(ul, ... , tin, E) _
+oo
Ean,(u1 i ... , un)E' of Gevrey order s as a -' 0 in D(r, a, b) uniformly
mc0
on D(rl,... , r.), where the coefficients a,n(u1 i ... , un) are holomorphic in
D(ri, . . . , rn),
(iii)n C-valued functions 61 (v", e), .... On(v, e) are holomorphic in a domain D x D(r, a, b), where D is a domain in the 6-space,
(iv)for each j, the function 03 (v, e) admits an asymptotic expansion p, (v, e) _
+oo
E b3,,n(v')e'n of Gevrey order si as a -+ 0 in D(r, a, b) uniformly on V, where
m=1
the coefficients bj,,n(v) are holomorphic in D.
Regarding the coefficients a,n(u1,... , un) as power series in ( u1 ,
, u.), define a
formal power series in a by
P(16, c) = p(p1(i', e)....
Pi-(Of"
M=0
where the coefficients P,n(v") are holomorphic in D. Then, for any (a, 0) satisfying the condition a < a < Q < b, there exists a positive number p(a, 3) such that
0 < p(a, (3) < r and that the function F(Oi (6, e), ... , thin (6,t), c) is holomorphic in
D x D(p(a, 0), a, Q) and admits the formal power series P(U, e) as its asymptotic expansion of Gevrey order max(s, si , ... , sn) as e 0 in V(p(a, 03), a, 03) uniformly on the domain D.
Remark XI-4-2. As a consequence of this theorem. we conclude that under the same assumptions as in Theorem XI-4-1, the formal power series P(v, e) in a is of Gevrey order max(s, s1, ... , sn) uniformly on D.
Proof of Theorem XI-4-1.
Fixing a pair (a, (3) satisfying the condition a < a < 0 < b, choose a suitable good covering {S1, ... , SN } ate = 0 and N(n + 1) functions Ft(u1, ... , un, e),
¢1,e(vU, e), ... , QSn,t(U, e) (e = 1,... , N) such that
(1)for each e, the function Ft(ui,... ,un,e) is holomorphic in the domain
D(r1 i ... , r,,) x St and admits the formal power series p(u1,... , Unit) as its asymptotic expansion of Gevrey order s as e -' 0 in St uniformly on
D(r1,... , rn),
(2)for each (j, e), the function 6,,e(17, e) is holomorphic in the domain V x St and
admits the formal power series pj (u", e) as its asymptotic expansion of Gevrey order sJ as e -+ 0 in St uniformly on V.
(3) there exist two positive numbers K and A such that
a
,un,E) - Ft+1(ul,... Un,()! < Kexp - Iefk
362 XI. ASYMPTOTICS EXPANSIONS
on D(rl,... , rn) x (St n St+l ), where k = 1,
(4) for each j = 1, ... , n, it holds that
101,1 (V,e) -o,,e+1(V,e)I Kexp -
k
on
D x (St n St+1),
JEI
where k, = 1 ,
(5)
Si
n) for some fo.
$t = D(p, a, E3), Ft. = F, and O,,&
We can accomplish this by virtue of Corollary XI-2-5 and Theorem XI-3-2.
for (6, e) E V x (St n S[+1), where L, K, and A are suitable positive numbers.
Therefore, using Theorem XI-2-3, we complete the proof of Theorem XI-4-1. 0
As a corollary of Theorem X1-4-1, the following result is obtained without any complication.
Theorem XI-4-3. C([x]], and A. (r, a, b) are commutative differential algebra over C, t.e.,
(i) f + g E C((x]], (respectively A. (r, a, b)) if f and g E C((x]], (respectively
A, (r, a, b)),
(ii) f9 E C([xJ], (respectively A. (r, a, b)) if f and g E C((xi], (respectively
A.(r, a, b)),
(iii)cf E C[(x]], (respectively A. (r, a, b)) if c E C and f E C((x]J, (respectively
A.(r, a, b)),
(iv)1 E C((xJ], (respectively A, (r, a. b)) if f E C((x)J, (respectively A. (r, a, b)),
(v) / Z f dx E C((x]], (respectively A, (r, a, b)) if f E C((xf, (respectively
A. (r, a, b))
Furthermore, the map J : A. (r, a, b) --t C((xi], is a homomorphism of differential algebra over C.
Remark XI-4-4. Using Theorem XI-3-6, we can prove (iv) and (v) of Theorem XI-4-3 in a way similar to the proof of Theorem X1-4-1. Also, it can be shown
that if f (x) E A, (r, a, b) with J(f) = >+"Qa,,,x' and ao 54 0, we obtain 1 E
m=o
f
5. PROOF OF LEMMA XI-2-6
363
A. (p(a, 8), a, 0) and J
\ I
/
= J{f) , where a < a < /3 < b and p(a, /3) is a
suitable positive number depending on (a,fl).
The materials in this section are also found in [Ram1], [Ram2], and [Si17, Ap- pendices].
XI-5. Proof of Lemma XI-2-6
In order to prove Lemma XI-2-6, choose N line segments C1, C2, ... , CN so that Ct E St n St+1 (t = 1, 2,... , N), i.e., Ct : z = teie1 (0 < t < r), where for each t, Bt is a fixed number satisfying the condition at+1 < 8t < bt. These N line segments divide the open punctured disk V = {x : 0 < IzI < r} into N open
sectors S1, S2, ... , SN, where
St = {x : 9t_ 1 < arg(x) < 01, 0 < txI < r}.
Set
N
Zt(x) =
21rt
lr
1
h=1
for x E St, t = 1, 2,... , N. The functions +Gt can be continued analytically onto
St by deforming Ct_1 and Ct without moving any of their endpoints. In doing this, we do not change other line segments Ch (h i4 t - 1, t). Thus, the function tyt is holomorphic on St, t = 1, 2,... , N, respectively.
Now, assuming that x E St n St+1, compute t/t(x) - tPt+1(x). To do this, write
'Pt and tLt+1 in the following forms respectively:
tt't(x) _
{2 =)
/ ft
(2 =)
E C"
Ln2
h#lJ
Ot+1 {x) =
27ri
e", t - x
+ 2Ri
c, - x
where the paths Ct and Ct+1 of integration are obtained by deforming C, without moving either of its endpoints so that
(1)Ct C St n St+l and Ct+l c St n St+1,
(2)-Ct + Ct+1 is a simple closed curve whose interior contains x.
Thus, (b) follows, i.e.,
0&) - 01+1(x) = tai fe, at(x) on St nSt,.1.
+ e«,
Let us derive asymptotic properties of ib1. To do this, fix an t and a closed sector
W contained in St. Let Ct (respectively Ct_1) be a path obtained by deforming Ct (respectively CL_1) without moving the endpoints so that W is contained in the
364 XI. ASYMPTOTIC EXPANSIONS
interior of the simple closed curve Ct_ 1 + 7t - C1, where 1't is a circular arc joining two points reist-' and rest. For x E W, it holds that
= {-1)
f
sew
(-1)
f
bt-1(1) d{
(-1)
f
x
ot (x) - 27ri
x
+ 2?7
x
+ 27ri
It may be assumed that the path Ct is given as a union of a line segment L1 and a curve rt defined by
Lt : x = te'"
(0 < t < r1 < r),
rt : x = Ftt(T)
(0 <T < 1),
where we is a constant such that 01 < we < bt, and
It can be also assumed that there exists a positive number a < 1 such that JxI < ar1
for x E W. Then, 1 8t(-)dC is represented by a convergent power series in x
21ri fr, c - x
whose radius of convergence is not less than r1.
Let us estimate the integral 1
bt()
Since
27ri
JL, c - x
1
mM+1
x,
x
x
Q)m +
1
1
we obtain
M
1
1
6t{f)
xM+lEM+1(x),
[Ymx"'
27ri
L - x
m=0
where
am
__ 1 r 5()
(m > 0)
and
EAi+I(x) =
1 f
61W
<.
21ri JL, Cm+1
-
27ri
£M+1({ - x)
Now, estimate the a,,, as follows:
=
1
I fr'
ebt(Teu.")eudJ
I
/r'
lQm
J
27r
f
Tm+1 dT
7
f
+oo
r_m-1e_ar-
dr =
7
A-Ilk m r(
l
27r
2k-,r l
!
\ k 1
-I-
\A-1lk\m
27)1-k (v') -m/k (m!)l",
(
\
7r
where the Stirling's formula (XI.3.3) was used as m -' +co.
EXERCISES XI
365
To estimate EN+i(x), note that there exists a positive number 0 depending on
W such that
t E L1, x E W.
it - xJ > Jtj sin(0)
for
Hence,
JEN+i(x)J
1 r
^ye-ar
'
<
.ya-(N+1)/k
N+ 1
2a Jo
rN+2sin(0)dr
2kirsin(0) r (
k
7/N+llity+l)/k
< k sin(O) ` )eke )
We can estimate
-11
2ri c,_, f - x
and
-1
(h94 Q,t-1)
2iri Jc - x
in a similar manner. Thus, the proof of Lemma XI-2-6 is completed.
The material of this section is also found in [Si18].
EXERCISES XI
XI-1. Let S be an open sector D(ro, a, b), i.e.,
S = {x : 0 < JxI < ro, a < argx < b},
where a and b are real numbers and ro is a positive number. Denote by
(a)A(S) the set of all functions which are holomorphic in S and admit asymptotic expansions in powers of x as x -. 0 in every open subsector
W = {x : 0 < JxJ < r < ro, a < a < argx < Q < b}
of S,
(b) J[f] the asymptotic expansion for f E A(S),
(c) Ao(S) the set of all those functions f (x) in A(S) such that f (x) ^-- 0 as x 0 in every W (i.e. AO(S) = If E A(S) : J[f] = 0}).
Show that C[Jxll. A(S), and Ao(S) are differential algebras over the field C and
that the map J : A(S)
C[[x] is a homomorphism of differential algebras over
the field C.
XI-2. Using the same notations as in Exercise XI-1 and considering a linear dif-
N
ferential operator P = E ak(x)Dk with coefficients ak(x) which are holomorphic k-o
in a disk JxJ < r containing S, where D = d, show that
366
XI. ASYMPTOTIC EXPANSIONS
(i) P defines three homomorphisms
P1 = P : Ao(S) - Ao(S), P2 = P : A(S) -- A(S), P3 = P : C[[x]] C[[x]]
of vector spaces over the field C,
(ii) two vector spaces A(S)/Ao(S) and C[[x]] are isomorphic,
(iii) dimensions of three vector spaces Nullspace ofP1i Nullspace ofP2, and
Nullspace ofP3 over C are not greater than the order of P,
(iv) two vector spaces {Nullspace of P2}/{Nullspace of P1} and J(Nullspace of P2] are isomorphic,
(v) there exists a homomorphism T : Nullspace of P3 .A0(S)/P1 [A0(S)] of vec- tor spaces over C such that Nullspace of T = J[Nullspace of P2] and
T[Nullspace of P3] = {P2[A(S)] nAo(S)}/P1[Ao(S)]
Hint.
(ii)J is onto (cf. Theorem X1-1-13) and .Ao(S) is Nullspace of J.
(iii)Use Theorem IV-2-1 for Nullspace of P1 and Nullspace of P2; use an idea similar to the proof of Theorem V-I-3 for Nullspace of P3.
(iv)Nullspace of the homomorphism J : Nullspace of P2 -+ J[Nulspace of P2] is
Nullspace of P1. Also, this homomorphism is onto.
(v)Assume that P3[¢] = 0 in C{[xJI. There exists a b E A(S) such that J[4] = 4
(cf. Theorem XI-1-13). Such b is not unique. Actually, ¢ + also satisfies the condition J[4, + '] = 4, if and only if iP E A0(S). So define T by T[4,] = P2[4J (mod P1[Ao(S)]). Note that J[P2[4,]] = P3[4'] = 0. This implies that
P2[O] E Ao(S).
X1-3. For each of the following three power series, find s such that f E C[[x]],.
cc
00 r(1+
2 )
(a) f(x) _
(mx)m,
m
(b) f (x) = E
3
,n=o
m_or 1 +
002m
(c)f (x) _ Exm 11 (5+ V5-h).
m=o h=m
XI-4. For two real numbers s > 0 and A > 0 and for a formal power series
+00
+00
f = 1: a,nxm E C[[x]1, set I I f 11 &,A = E (
Denote by E(s, A) the set of
M=0
all f E7=0C[[x]] satisfying the condition IIf II..A < +oc. Show that
(i) E(s, A) is a Banach space with the norm IIf II.,A,
(ii)
IIf9II.,A <_ IIf II..A I19II..A if f E E(s, A) and g E E(s, A),
(iii)
if B > A, then E(s, A) C E(s, B), and II f II,,a < IIf II,,A for f E E(s, A),
(iv) if B > A, then Il f II.,e - If (0)I <
[IIf II.,A - If (0)I] for f E .6(s, A), where
f (0) = a0 if f = E.m>o a. xm,
B
(v) for f E E(s, A) and an integer q such that sq > 1, it holds that
df
5 -LIIfII.,A
11
eA
EXERCISES XI
367
Comment. The formal power series f E C([x]] is of Gevrey order s if and only if f E E(s, A) for some positive number A.
XI-5. Show that f (e) = 0 identically in a sector S if f is holomorphic and flat of
Gevrey order 0 as e --+ 0 in S.
Hint. I f (e)I < KA' IeIr' fore E Sand N = 0,1,2,..., where K and A are positive numbers independent of N.
XI-6. Show that C[[x]]o = C{x}.
Hint. f = 00E
E C[[x]]o if and only if Ic,.. I < KAt If I' (m = 0, 1, 2.... ),
=G
where K andmA are positive numbers independent of m.
X1-7. Consider a system of differential equations
(S)
xy+t dy
= f0 + bg +
Ip!>_o
where q is a positive integer, fo and fp E C[(x]]n, 4> is an n x n matrix whose entries are formal power series in x, p = (pl, ... , p,>) with n non-negative integers pt, ... ,
Pn, IpI = Pt + - - - + p,,, and yam' = yP' - - - VP-.Assume that sq > 1 and that
(i)fo and f. E E(s, A)n, and the entries of 0 belongs to E(s, A) for some A > 0,
(ii)the power series F, IIfpIIs,A9P is a convergent power series in y", where
Ipl>2
IIY1Ia,A = max{IIy,II5,A :1 < j < n},
(iii)fo(0) = 0,
(iv)(3(0) is invertible.
Show that there exists a unique power series E C[[x]]n satisfying system (S) and the condition ¢(0) = 0. Furthermore, E (C[[x]],)n.
Hint. Write system (S) in a form y" = go + x9+1 1P + E y'gp and use the
ipl>2
Banach fixed-point theorem in terms of the norm II dx
IIs.A-
XI-8. Let D(r) = {z : IzI < r}, S(r, p) = {z : 0 < IzI < r, I arg zI < p}, and a power
+w
series f (z, e) = E fn(e)zn is convergent uniformly in a domain D(ro) x S(r, p),
n=o
where ro, r, and p are positive numbers. Assume that the coefficients fn(e) of the series f are holomorphic in S(r, p) and admit asymptotic expansions in powers of
e as e -' 0 in S(r, p). Suppose also that limo fo(e) = 0 and limo f, (c) 0 0. Show
C-
1-
that there exist a positive number rt and a function O(e) such that (1) O(e) is holomorphic in S(rt, p) and admits an asymptotic expansion in powers of a as e -+ 0 in S(rt, p), (2) l ¢(e) = 0, and (3) f (c(e), e) = 0 identically in S(rt, p).
