8.2 More on Transformation to Rational Form

In Theorem 11 (p. 52) we showed that a formal system can be transformed

into a rational one “ meaning a system with a coe¬cient matrix being a ra-

tional function with poles at the origin and in¬nity only. Here we shall prove

that, starting with a system (3.3) whose coe¬cient matrix is r-summable,

then so is the transformation obtained in the said theorem. Indeed, much

more can be said:

ˆ ˆ

Proposition 19 For some d ∈ R, let zx = A(z) x and z x = B(z) x be

˜ ˜

two formal system of Poincar´ rank r ≥ 1 whose coe¬cient matrices are

e

ˆ

r-summable in some direction d, and let T (z) be a formal meromorphic

transformation of Gevrey order s = 1/r satisfying (3.5). Furthermore, as-

sume that ’rd = arg(µ ’ µ) mod 2π, for any two distinct eigenvalues µ, µ

˜ ˜

ˆ(z) is also r-summable in direction d.

of the leading term A0 . Then T

ˆ

Proof: Using Exercise 4 on p. 41, applied to the transpose of T (z), we factor

ˆ ˆ

T (z) = T0 (z) T (z) with a terminating meromorphic transformation T (z)

ˆ

and a formal analytic transformation T0 (z) of Gevrey order s and leading

term I. Setting B0 (z) = (T (z)B(z) + zT (z))T ’1 (z), we see that B0 (z)

ˆ ˆ ˆ

ˆ

is also r-summable in direction d, while T (z) is r-summable in direction

ˆ

d if and only if the same holds for T0 (z). This shows that in the proof

ˆ

we may without loss of generality assume that T (z) is a formal analytic

ˆ

transformation with constant term I, and then the leading term of B(z) is

equal to A0 , and its Poincar´ rank is again equal to r.

e

Using the same notation as in the proof of Theorem 11 (p. 52), we con-

clude from our assumptions that the matrices ur’1 A(u) and ur’1 B(u)

are holomorphic in G = D(0, ρ) ∪ S(’d, µ), for su¬ciently small ρ, µ > 0,

and of exponential growth at most r there. Moreover, we conclude that

T (u) is holomorphic in D(0, ρ), with ρ small enough, except for a pole of

order at most r ’ 1, and satis¬es the integral equation (3.17) (p. 53) for

|u| < ρ. Making the above µ > 0 smaller if necessary, we can arrange that

the equation rur = µ ’ µ, with µ, µ as above, does not have a solution in

˜ ˜

128 8. Solutions of Highest Level

G. Then this integral equation, according to the exercises below, implies

that T (u) can be holomorphically continued into G and is of exponential

growth at most k there. This, however, is equivalent to k-summability of

ˆ 2

T (z) in direction d.

It is worthwhile emphasizing that the integral equation (3.17) (p. 53),

under the assumptions stated above, in general is singular at the origin,

meaning roughly that one cannot use the standard iteration procedure to

show existence of a solution near the origin. However, once a solution is

known, the integral equation can be used for holomorphic continuation, and

a growth estimate for the solution follows from analogous estimates of the

terms in the equation.

In the construction of HLFFS, Theorem 11 (p. 52) is applied to the

diagonal blocks of a system obtained by an application of the Splitting

Lemma. These blocks have been shown above to be r-summable, and by

construction their leading term only has one eigenvalue. In this situation,

the above proposition shows that no additional singular rays occur, because

the condition in terms of the eigenvalues of A0 is void.

Exercises: In the following exercises, use the same notation and assump-

tions as in the proof of the above proposition.

1. For ¬xed u0 = x0 ei„ with |d + „ | < µ, assume that T (u) has been

holomorphically continued along the line segment u = xei„ , 0 ¤ x ¤

x0 , which trivially holds for x0 < ρ. De¬ne C(u) = A(u) ’ B(u) +

u0

[ A([ur ’ tr ]1/r ) T (t) ’ T (t) B([ur ’ tr ]1/r ) ] dtr , and show:

0

(a) The function C(u) is holomorphic in a small region Gu0 ,δ = {u :

|„ ’ arg(u ’ u0 )| < δ}, δ > 0.

(b) The integral equation T (u) A0 ’ (A0 + rur I) T (u) = C(u) +

u

[ A([ur ’ tr ]1/r ) T (t) ’ T (t) B([ur ’ tr ]1/r ) ] dtr has a unique

u0

solution T (u) that is holomorphic in Gu0 ,δ and of exponential

growth at most r.

2. Show that the function T (u) is holomorphic in G and of exponential

growth not more than r there.

3. Prove the following Corollary to Proposition 19: In addition to

ˆ ˆ

the assumptions of Proposition 19, let A(z) and B(z) be convergent,

ˆ

and let A0 have only one eigenvalue. Then the transformation T (z)

converges as well.

8.3 Summability of Highest-Level Formal Solutions 129

8.3 Summability of Highest-Level Formal Solutions

On p. 55 we have given the de¬nition of HLFFS and its data pairs. Using the

results of the previous sections, we are now ready to investigate summability

of HLFFS. As will become clear later, the following theorem, in a way, is

the main one in our theory of HLFFS:

Theorem 42 (Main Theorem) Assume that we are given a system (3.1)

(p. 37) having an essentially irregular singularity at in¬nity. Then the fol-

lowing holds true:

(a) For any two HLFFS of (3.1), the data p, q agree. In particular, both

HLFFS are of the same Gevrey order s = 1/k, k = r ’ p/q.

(b) For any two HLFFS of (3.1), the data pairs (q1 (z), s1 ), . . . , (qµ (z), sµ )

agree modulo renumeration, so in particular the number µ of such

pairs is the same.

ˆ ˆ

(c) For any two HLFFS F1 (z), F2 (z), assume that their data pairs co-

incide, which can always be brought about by suitably permuting the

columns of any one of the HLFFS. Then the transformation T (z) =

ˆ ’1 ˆ

F1 (z) F2 (z) is diagonally blocked of type (s1 , . . . , sµ ) and converges.

(d) Every HLFFS is k-summable in all directions d, except for

(p)

’kd = arg(»j ’ »(p) ) mod 2π, j = m, 1 ¤ j, m ¤ µ. (8.3)

m

ˆ

Proof: Let Fj (z), 1 ¤ j ¤ 2, be any two HLFFS of (3.1), and let Bj (z)

be the coe¬cient matrices of the corresponding transformed systems, as

ˆ ’1

ˆ ˆ

de¬ned on p. 55. Setting T (z) = F1 (z) F2 (z), one can show formally

ˆ ˆ ˆ

z T (z) = B1 (z) T (z) ’ T (z) B2 (z). (8.4)

Let q = m1 q1 = m2 q2 be the least common multiple of q1 , q2 . Origi-

nally, the matrix Bj (z) has been expanded with respect to the variable

wj = z 1/qj , and the ¬rst pj coe¬cients are scalar multiples of the unit

matrix. Re-expanding Bj (z) in the new variable w = z 1/q , the ¬rst mj pj

coe¬cients now are scalar, and some may even be zero. Assume for the

moment that both pj are positive, so that scalar coe¬cients occur. Make

the change of variable z = wq in (8.4), and then apply Exercise 2, with

ˆ ˆ

µ = µ = 1, i.e., A(z) = B1 (z q ), B(z) = B2 (z q ) blocked trivially. This

˜

implies that the highest terms of both Bj (z) are the same scalar multiple

ˆ

of the identity matrix. Since these terms commute with T (z), they can be

cancelled from (8.4). Then, the same argument can be used over again,

until one of the highest terms, after cancellation of the ones shown to be

identical, is no longer scalar. In this situation, an application of Exercise 2

130 8. Solutions of Highest Level

implies that then both leading terms must have more than one eigenvalue;

hence m1 p1 = m2 p2 follows. Together with q = m1 q1 = m2 q2 , this implies

p1 /q1 = p2 /q2 . By assumption pj , qj are co-prime except for qj = 1; so we

conclude q1 = q2 and p1 = p2 . This shows (a). However, applying Exer-

cise 2 to the situation where both leading terms have several eigenvalues,

we even obtain statements (b) and (c).

To show (d), observe that according to (c), it su¬ces to show the ex-

istence of one HLFFS that is summable as stated. This, however, is a

consequence of Lemma 11 (p. 124) and Proposition 19 (p. 127), combined

2

with the algorithm of constructing an HLFFS as stated on p. 50.

The above Main Theorem not only implies k-summability of HLFFS. It

also supplies information on the degree of freedom in these objects:

On one hand, it is shown that the data pairs, up to enumeration, cor-

respond uniquely to a given system (3.1), and to every enumeration of the

data pairs one can ¬nd an HLFFS, simply by a suitable permutation of

ˆ

its columns. Moreover, the q-meromorphic transformation F (z) is de¬ned

up to a left-hand-side factor T (z) that is a diagonally blocked convergent

q-meromorphic transformation. This transformation, owing to the de¬ni-

tion of HLFFS, transforms one system z y = B(z) y into another one, say,

˜

z y = B(z) y , and both coe¬cient matrices have terminating expansions

˜ ˜

in the variable w = z 1/q . It is clear that, given one HLFFS, multiplication

from the left with such a factor T (z) again gives an HLFFS. Unfortunately,

it is not at all obvious which diagonally blocked q-meromorphic transfor-

mations, when used to transform a system z y = B(z) y with terminating

expansion, will produce another one having the same property. This, how-

ever, will not be of importance for us.

Exercises:

1. Let An ∈ C ν—ν , Bn ∈ C ν —˜ , with r ≥ 0, ν, ν ≥ 1 not necessarily

˜ν

˜

equal, and n ≥ 0. Let Tn ∈ C , n ≥ n0 (n0 ∈ Z) satisfy

ν—˜

ν

n

’(n ’ r) Tn’r = (An’m Tn ’ Tn Bn’m ) , n ≥ n0 ,

m=n0

with Tn = 0 for n < n0 . If r ≥ 1 and A0 , B0 have disjoint spectrum,

show that Tn = 0 for every n ≥ n0 follows. For r = 0, ¬nd a necessary

and su¬cient condition in terms of the spectra of A0 , B0 under which

the same conclusion holds.

ˆ

2. Let a formal meromorphic transformation T (z) satisfy (3.5) (p. 40),

ˆ ˆ

with matrices A(z), B(z) of the following form:

(a) The matrices z ’r A(z) and z ’r B(z) are formal matrix power

ˆ ˆ

’1

series in z with constant terms A0 resp. B0 that may vanish

8.4 Factorization of Formal Fundamental Solutions 131

or not (in other words: the corresponding formal systems have

Poincar´ ranks at most r), and both A0 and B0 are in Jordan

e

form, say,

A0 = diag [»1 Is1 + N1 , . . . , »µ Isµ + Nµ ],

˜˜ ˜˜ ˜

˜ ˜˜

B0 = diag [»1 Is1 + N1 , . . . , »µ Isµ + Nµ ],

˜

˜ ˜˜

with distinct »1 , . . . , »µ , resp. »1 , . . . , »µ and nilpotent matrices

˜

Nj , resp. Nj .

ˆ ˆ

(b) The matrices A(z) and B(z) are diagonally blocked in the block

structures of their leading terms A0 , resp. B0 .

˜˜

Show that then µ = µ, and the pairs (»j , sj ) and (»j , sj ) agree up

˜

ˆ

to a renumeration. In case the pairs are identical, show that T (z) is

ˆ ˆ

diagonally blocked, and converges if both A(z) and B(z) converge.

3. Under the assumptions of the previous exercise, assume that the pairs

˜˜

(»j , sj ) and (»j , sj ) are not identical. Show that then one can permute

ˆ

the columns, resp. the rows, of T (z) so that the resulting matrix is

ˆ ˆ

diagonally blocked in the block structure of A(z), resp. of B(z).

8.4 Factorization of Formal Fundamental Solutions

We now are going to discuss brie¬‚y the classical notion of formal funda-

mental solutions (FFS for short) of a system (3.1) (p. 37). Such a FFS of

a convergent system will, in general, not be k-summable for any k > 0.

However, we are going to show that it can always be factored into ¬nitely

many terms that individually are k-summable, but for values k depending

on the factor.

In principle, one can compute a formal fundamental solution even for for-

mal systems (3.3) (p. 40). To do so is equivalent to ¬nding a q-meromorphic

ˆ

formal transformation F (z) producing an elementary formal transformed

∞

system, say, with coe¬cient matrix B(z) = wr 0 Bn w’n , w = z 1/q ,

˜

ˆ

where all Bn commute with one another. It follows from the exercises be-

low that then another q-meromorphic transformation takes (3.3) into