one can therefore characterize the image of the operator Sk,d . This will be

done in detail in the context of multisummable series later on.

Exercises: Throughout the following exercises, let » be any complex

number, let p(z) be a polynomial of degree r ≥ 1 and highest coe¬cient

one, and let g(u) be holomorphic (and single-valued) for 0 ¤ |u| < ρ.

1. For j = 0, . . . , r, de¬ne

z

u’» e’p(u) g(1/u) dur , 0 < |z| < ρ,

» p(z)

fj (1/z) = z e

∞(2jπ/r)

integrating from ∞ along the line arg u = 2jπ/r to some arbitrarily

chosen point z0,j , |z0,j |’1 < ρ, and then to z. Show that each fj is

holomorphic for 0 < |z| < ρ, and fj (1/z) ’ fj’1 (1/z) = cj z » ep(z) ,

j = 1, . . . , r, with cj = γj u’» e’p(u) g(1/u) du, where γj is a path

from ∞ along arg u = 2jπ/r to z0,j , then to z0,j’1 , and back to ∞

along arg u = 2(j ’ 1)π/r.

2. For z ∈ Sj = S(2jπ/r, 3π/r, ρ) and j = 0, . . . , r, show that fj (z) is

bounded at the origin.

3. Show

fj’1 (z) ’ fj (z) ∈ A1/r,0 (Sj’1 © Sj , E ), 1 ¤ j ¤ r,

fr (z) = f0 (ze’2πi ), z ∈ Sr .

4. Conclude fj ∈ A1/r (Sj , E ), 0 ¤ j ¤ r.

8

Solutions of Highest Level

In this chapter we are going to prove that the formal transformations oc-

curring in the Splitting Lemma (p. 42) and in Theorem 11 (p. 52) are, in

fact, r-summable in the sense of Chapter 6. Based upon this, we shall then

show that highest-level formal fundamental solutions are k-summable for

1/k = s equal to their Gevrey order. As an application of this result we

then prove the factorization of formal fundamental solutions according to

Ramis and the author, as described in Chapter 14. For every highest-level

formal fundamental solution (HLFFS for short), we then de¬ne normal so-

lutions of highest level that were ¬rst introduced by the author in [8], and

we shall investigate their properties in more detail, de¬ning corresponding

Stokes™ directions, Stokes™ multipliers, etc.

Since in Chapter 3 all power series have been in the variable 1/z, we

have to make some more or less obvious adjustments in the de¬nition of

r-summability. In particular, we de¬ne sectorial regions at in¬nity to be

such regions G that by the inversion z ’ 1/z are mapped onto sectorial

ˆ

regions in the previous sense, and from now on, we shall denote by f (z) a

formal power series in the variable z ’1 . It is then natural to say that

• a power series f (z) ∈ E [[z ’1 ]] is k-summable in direction d if and

ˆ

∞

only if f (z ’1 ) = 0 fn z n is k-summable in direction ’d.

ˆ

The formal Borel transform Bk f is de¬ned by applying Bk to f (z ’1 ), and

ˆˆ ˆ

ˆ

its sum then is holomorphic in a small sector of bisecting direction ’d, and

ˆ

is of exponential growth at most k there. The k-sum of f (z) then is obtained

by summing f (z ’1 ) followed by the change of variable z ’ 1/z. Hence, if

ˆ

124 8. Solutions of Highest Level

ˆ

f (z) is the k-sum in direction d of a power series f (z) in inverse powers

of z, then f (z) is holomorphic in a sectorial region G(d, ±) (near in¬nity)

of bisecting direction d and opening ± > π/k, and f (z) can be represented

by a Laplace integral (5.1), with z replaced by 1/z and the direction of

integration „ close to ’d. So here, turning the path of integration in the

positive sense leads to holomorphic continuation of f (z) in the negative

sense, and vice versa.

Because HLFFS in general are series in some root of z ’1 , we need to

generalize the notion of k-summability to q-meromorphic transformations,

or more generally to arbitrary formal Laurent series in the variable z ’1/q :

• A power series in z ’1/q will be called k-summable in direction d, if the

series obtained by the change of variable z = wq is qk-summable in

direction d/q. Compare Exercise 2 on p. 72 to see that this de¬nition

gives good sense even if the series we start with accidentally happens

to not contain any roots.

∞

• A formal Laurent series n=’m fn z ’n is called k-summable in di-

∞

rection d if and only if its power series part f (z) = n=0 fn z ’n is

ˆ

summable in this sense, with the k-sum of the Laurent series equal

to the k-sum of the power series part plus the ¬nite principle part

’1 ’n

n=’m fn z . Compare this to Exercise 5 on p. 72.

8.1 The Improved Splitting Lemma

In the construction of HLFFS we had to consider formal analytic transfor-

mations in two places: In the Splitting Lemma in Section 3.2, and later in

Section 3.4, ¬nding a transformation so that the new system is rational.

To proceed, we ¬rst reconsider the proof of the Splitting Lemma and show

that, beginning with a convergent system (3.1), the transformation as well

as the transformed system are r-summable, and we can even completely

describe the set of singular directions.

Lemma 11 (Improved Splitting Lemma) Consider a convergent sys-

tem (3.1) (p. 37) satisfying the assumptions of the Splitting Lemma on p. 42.

ˆ ˆ

Then the formal matrix power series T12 (z) and the block B22 (z) of the coef-

¬cient matrix of the transformed system are r-summable in every direction

d except for the ¬nitely many singular ones of the form ’rd = arg(µ1 ’ µ2 )

(jj)

mod 2π, with µj an eigenvalue of A0 . The same statement, but with µ1 , µ2

ˆ ˆ

interchanged, holds for T21 (z) and B11 (z).

Proof: Analogously to the proof of Theorem 11 (p. 52), we de¬ne T (u) =

∞ ∞

n’r n’r

1 T12 u /“(n/r), B(u) = 1 B22 u /“(n/r), and (slightly abus-

∞ (jk) n’r

/“(n/r). Then ur’1 Ajk (u) are en-

ing notation) Ajk (u) = 1 An u

8.1 The Improved Splitting Lemma 125

tire functions of exponential growth at most r, owing to the convergence

of (3.1). Moreover, (3.6) and (3.7) (p. 42) are formally equivalent to the

integral equation

(22) (11)

’ (A0 + rur I) T (u)

T (u) A0 =

u

A11 ([ur ’ tr ]1/r ) T (t) ’ T (t) B([ur ’ tr ]1/r ) dtr ,

A12 (u) + (8.1)

0

with u

A21 ([ur ’ tr ]1/r ) T (t) dtr .

B(u) = A22 (u) + (8.2)

0

The Splitting Lemma ensures existence of T (u) and B(u), holomorphic and

single-valued near the origin and satisfying (8.1). We aim at proving that

both can be holomorphically continued into the largest star-shaped1 region

(22) (11)

G that does not contain any point u for which A0 and A0 + rur I have

an eigenvalue in common. Observe Exercise 3 on p. 214 to see that this

is the largest star-shaped set not containing any solution of the equation

rur = µ2 ’ µ1 . Moreover, we shall show that in this region both matrices

have exponential growth at most r. This then implies r-summability of

ˆ ˆ

T12 (z) and B22 (z), with singular directions as stated, and one can argue

analogously for the other two blocks.

To do all this, we employ an iteration: Beginning with B(u; 0) ≡ 0,

T (u; 0) ≡ 0, we plug B(u; m) and T (u; m) into the right-hand sides of

(8.1), (8.2) and determine T (u; m + 1) from the left-hand side of (8.1),

resp. let B(u; m + 1) be equal to B(u) in (8.2). Both sequences so obtained

are holomorphic in G, except for a pole of order at most r ’ 1 at the

origin. For the following estimates, we choose d so that the ray u = xe’id ,

x ≥ 0, is in G, hence d is a non-singular direction. Then Ajk (u) ¤

∞ n n’r

/“(n/r), with a ≥ 0 independent of d. Inductively we show

1ax

∞ (m) n’r

estimates of the form T (u; m) ¤ /“(n/r), B(u; m) ¤

1 tn x

∞ (m) n’r (0) (0)

1 bn x /“(n/r): For m = 0 this is certainly correct with tn = bn =

0. Given this estimate for some m ≥ 0, we use the recursion formulas and

the Beta Integral (p. 229) to show the same type of estimate for m + 1. In

particular, we may set

(m+1) (m)

n’1

= an + 1 an’k tk

bn

n ≥ 1,

(m) (m)

(m+1) n’1 n’k

n

tn = c a + 1 (a + bn’k )tk

where c is a constant which arises when solving the left-hand side of (8.1)

for T (u) and then estimating the solution “ hence c is independent of n

and m, as well as independent of d provided we alter d only slightly so that

1 With respect to the origin.

126 8. Solutions of Highest Level

it keeps a positive distance from the singular directions. For every n, the

(m) (m)

numbers tn , bn can be seen to be monotonically increasing with respect

to m and become constant when m ≥ n. Their limiting values tn , bn then

satisfy the same recursion equations, with the superscripts dropped. Setting

∞ ∞ ∞

b(x) = 1 bn xn , t(x) = 1 tn xn , a(x) = 1 an xn = ax(1 ’ ax)’1 , we

obtain formally b(x) = a(x)[1+t(x)], t(x) = c[a(x)+(a(x)+b(x))t(x)]. This

can be turned into a quadratic equation for t(x), having one solution t1 (x)

that is a holomorphic function near the origin, while the other one has a

pole there. The coe¬cients of t1 (x) satisfy the same recursion formulas as

tn , so are in fact equal to tn . This implies that both tn and bn cannot grow

faster than some constant to the power n. From this fact we conclude that

r

T (u; m) and B(u; m) can be estimated by c x1’r eKx with suitably large

c, K independent of m. Therefore, the proof will be completed provided

that we show convergence of T (u; m) and B(u; m) as m ’ ∞. This can be

done by deriving estimates, similar to the ones above, for the di¬erences

T (u; m) ’ T (u; m ’ 1) and B(u; m) ’ B(u; m ’ 1) and turning the sequence

2

into a telescoping sum. For details, compare the exercises below.

It is important to note for later applications that the above lemma not

ˆ

only ensures r-summability of the transformation T (z), but also allows ex-

plicit computation of the possible singular directions in terms of parameters

of the system (3.1) (p. 37). As we shall make clear in Chapter 9, it may

happen that some, or all, of these directions are nonsingular, depending on

whether some entries in Stokes™ multipliers are zero. However, in generic

cases all these directions are singular!

Exercises: In the following exercises, make the same assumptions as in

the lemma above, and use the same notation as in its proof.

1. For m ≥ 1 and u as in the proof of the above lemma, show

∞

d(m) xn’r /“(n/r),

T (u; m) ’ T (u; m ’ 1) ¤ n

n=m

∞

d(m) xn’r /“(n/r),

˜

B(u; m) ’ B(u; m ’ 1) ¤ n

n=m

(m) ˜(m)

with constants dn , dn satisfying

n’1 n’1

(m) ˜(m) (m)

d(m+1) d(m+1) =

˜

n’k

an’k dk ,

= b (dk + dk ),

n n

k=m k=m

for suitably large b > 0, and a as in the proof of the lemma.

(m)

∞ ˜

2. Setting dm (x) = n=m dn xn , and analogously for dm (x), derive a

recursion for these functions “ in particular, conclude convergence of

the series for 0 < x < min(a, b).

8.2 More on Transformation to Rational Form 127

˜ ˜

3. Setting d(x) = m dm (x), d(x) = m dm (x), use the previous exer-

cise to show functional equations, and from these compute the func-

˜

tions in terms of d1 , d1 . In particular, show that both functions are

holomorphic near the origin. Use this to conclude existence of d so

(m) ˜(m)

that dn , dn ¤ dn for every n, m.

4. Use the previous exercises to show that T (u; m) and B(u; m) converge

uniformly on every compact subset of the region G de¬ned in the

proof of the above lemma.