(21)

+A(12) ’ Tµ An’µ’m Tm ,

n

µ=1 m=1

for n ≥ 1. According to our assumption and Lemma 24 (p. 212), these

recursions determine the Tn uniquely, so to prove existence we are left to

show convergence of the power series so obtained for T (z). To do this we

proceed analogously to the proof of Theorem 5 (p. 19): De¬ning 3

n’µ

n’1

’µ

K ’m tm n ≥ 1,

n

tn = c K 1+ K tµ 2 + ,

µ=1 m=1

for K > 1/ρ and su¬ciently large c > 0, we show by induction Tn ¤

∞

tn , n ≥ 1. Putting f (z) = n

1 tn (z/K) , one can formally obtain the

quadratic equation (1 ’ z) f (z) = zc [1 + 2f (z)] + c f 2 (z). Its solutions are

f± (z) = (2c)’1 1 ’ z(2c + 1) ± [1 ’ z(2c + 1)]2 ’ 4c2 z ,

which both are holomorphic near the origin. Moreover, f’ (0) = 0, and

the coe¬cients of its power series expansion satisfy the same recursion as,

so are in fact equal to, the numbers tn K ’n . This proves convergence of

∞

1 tn z , for su¬ciently small |z|. 2

n

Note that, unlike for linear systems, we have not shown that the matrix

T (z) in the above lemma is holomorphic in the same disc as the coe¬cient

functions; in general, this is not true for nonlinear systems, as one can learn

from simple examples.

Lemma 2 may be rephrased as showing existence of a unique analytic

transformation of the form

I T12 (z)

T (z) = , T12 (0) = 0,

0 I

transforming (2.1) into (2.12), with

B11 (z) 0

B(z) = ,

B21 (z) B22 (z)

3 Observe that the same recursion, but with c replaced by c/n, would also ensure

Tn ¤ tn , and this majorizing sequence would clearly give a better estimate. However,

then the corresponding function f (z) is a solution of a Riccati di¬erential equation,

instead of a simple quadratic equation. Hence proving analyticity is more complicated.

30 2. Singularities of First Kind

B11 (z) = A11 (z) ’ T12 (z) A21 (z), B21 (z) = A21 (z),

B22 (z) = A21 (z) T12 (z) + A22 (z).

Iterating this result, we are able to prove that a system with general spec-

trum can be transformed into another one that is reduced in a block struc-

ture determined by the Jordan canonical form of A0 :

Proposition 3 Let a system (2.1) with general spectrum be given. As-

sume that the matrix A0 is diagonally blocked, with the diagonal blocks

having exactly one eigenvalue, and let these eigenvalues have increasing

real parts. Then an analytic transformation T (z) exists that is upper tri-

angularly blocked with respect to the block structure of A0 , with diagonal

blocks identically equal to I, so that the corresponding transformed system

is lower triangularly reduced in the same block structure.

Proof: We proceed by induction with respect to the number of blocks of

A0 . In the case of one such block nothing remains to prove, while for two

blocks Lemma 2 assures the existence of the transformation as stated. If

we have even more diagonal blocks in A0 , we block all matrices in a coarser

block structure with two diagonal blocks by grouping several of the blocks

of A0 into one large block. Then we can again apply Lemma 2 to obtain

a system that is lower triangularly blocked of this coarser type, with two

diagonal blocks to which the induction hypothesis applies. Since diagonally

blocked analytic transformations leave the system in lower triangularly

blocked form, we can use such a transformation to put all the diagonal

2

blocks into the desired form. This, however, completes the proof.

Note that the assumptions of the above proposition can always be made

to hold by a constant transformation T (z) ≡ T , putting A0 into Jordan

canonical form with the eigenvalues ordered accordingly. Hence, in princi-

ple the above proposition allows us to compute fundamental solutions of

systems with general spectrum: First, we ¬nd an analytic transformation

to another system that is reduced and has diagonal blocks, each of which

has good spectrum. Then, we compute a fundamental solution of the new,

reduced, system by means of Theorem 5, applied to each diagonal block,

and Theorem 4. This fundamental solution will be investigated further:

Proposition 4 Let a system (2.1) be given that is reduced of some type

(ν1 , . . . , νµ ), with Ajj (0) having exactly one eigenvalue »j , and assume that

these eigenvalues have increasing real parts. Moreover, let Akj (0) = 0 for

k > j. Finally, let κj ∈ Z be such that 0 ¤ Re »j ’ κj < 1. Then there

exists a triangularly blocked fundamental solution X(z) of (2.1) of the form

X(z) = T (z) z K z M ,

where T (z) is a lower triangularly blocked analytic transformation, M is

a lower triangularly blocked constant matrix with diagonal blocks equal to

Ajj (0) ’ κj I, and K = diag [κ1 Iν1 , . . . , κµ Iνµ ].

2.4 Systems with General Spectrum 31

Proof: Using Theorem 5 (p. 19), we compute fundamental solutions

Xjj (z) = Tjj (z) z Ajj (0) for the diagonal blocks of our system, and then

compute the blocks below the diagonal using Theorem 4 (p. 12), thus ob-

taining some lower triangularly blocked fundamental solution X(z). Obvi-

ously, X(z) has a monodromy factor that is blocked in the same way, and

according to Theorem 71 (p. 241), so is the unique monodromy matrix M

with eigenvalues in [0, 1). Its diagonal blocks Mjj are monodromy matrices

for the diagonal blocks of X(z), having eigenvalues with real parts in [0, 1).

Another such monodromy matrix is Ajj (0) ’ κj I; hence, according to the

uniqueness part of Theorem 71, we have Mjj = Ajj (0) ’ κj I. De¬ning

T (z) = X(z) z ’M z ’K , we conclude that T (z) is single-valued at the ori-

gin, and its diagonal blocks are even holomorphic there, while the others

could be singular there. Direct estimates of (1.15) (p. 13) show that the

blocks of T (z) below the diagonal cannot have an essential singularity at

the origin. Moreover, the de¬nition of T (z) implies

z T (z) = A(z) T (z) ’ T (z) B(z), (2.15)

with B(z) = K + z K M z ’K . Since κ1 , . . . , κµ are weakly increasing and M

is lower triangularly blocked, we conclude that B(z) is holomorphic at the

origin, and B(0) and A(0) have the same diagonal blocks. For the blocks

Tkj (z), 1 ¤ j < k ¤ ν, we obtain from (2.15) that

z Tkj (z) = Akk (z) Tkj (z) ’ Tkj (z) Bjj (z) + Rkj (z), (2.16)

with Rkj (z) depending only upon such blocks of T (z) that are closer to, or

on, the diagonal. Assume that we have shown Rkj (z) to be holomorphic at

the origin, which is correct for k ’ j = 1, since then Rkj (z) involves only

diagonal blocks of T (z). Then, let m ≥ 0 be the pole order of Tkj (z) and

(kj)

T’m denote the corresponding coe¬cient in its Laurent expansion. If m

(kj) (kj)

were positive, then (2.16) would imply [mI + Akk (0)] T’m = T’m Ajj (0).

(kj)

This, however, would imply T’m = 0, owing to our assumption on the

eigenvalues of the diagonal blocks. Hence m = 0 follows; i.e., Tkj (z) is

holomorphic at the origin. Therefore, by induction with respect to k ’ j we

2

¬nd that all blocks of T (z) must be holomorphic at the origin.

Together, the above two propositions clarify the structure of fundamental

solutions in case of general spectrum: First, we use a constant transforma-

tion to bring A0 into Jordan form, with eigenvalues ordered appropriately.

Then, we compute an upper-triangularly blocked analytic transformation,

such that the transformed equation is lower triangularly blocked. Finally,

we ¬nd a lower triangularly blocked analytic transformation for which the

transformed equation has an explicit solution z K z M . We state this main re-

sult of the current section in the following theorem, before we add a remark

concerning a more e¬cient way of computing this fundamental solution.

32 2. Singularities of First Kind

Theorem 6 A system (2.1) with general spectrum has a fundamental so-

lution of the form

X(z) = T (z) z K z M , (2.17)

where

• T (z) is an analytic transformation,

• M is constant, lower triangularly blocked of some type (ν1 , . . . , νµ ),

• the kth diagonal block of M has exactly one eigenvalue mk with real

part in the half-open interval [0, 1),

• K = diag [κ1 Iν1 , . . . , κµ Iνµ ], κj ∈ Z, weakly increasing and so that

mk + κk is an eigenvalue of A0 with algebraic multiplicity νk .

It is worth pointing out that the above result does not coincide with

Theorem 5 (p. 19) in case of a good spectrum, since the eigenvalues of A0

need not have real parts in [0, 1). Nonetheless, the theorem clari¬es the

structure of fundamental solutions of (2.1) with general spectrum, and the

propositions in principle allow the computation of the monodromy matrix

M , the diagonal matrix K, and any ¬nite number of coe¬cients of T (z).

For a more e¬ective computation, one may use the following procedure

from Gantmacher [105], which also provides another proof of Theorem 6:

∞

Remark 1: Let (2.1) and an analytic transformation T (z) = 0 Tn z n be

∞

arbitrarily given, and expand B(z) = 0 Bn z n . Then (2.13) is equivalent

to

n’1

Tn (B0 + nI) ’ A0 Tn = (An’m Tm ’ Tm Bn’m ), (2.18)

m=0

for n ≥ 0. We assume that A0 = diag [A11 , . . . , Aµµ ] is diagonally blocked as

in Proposition 3 (p. 30), which can always be brought about by a constant

(jk)

transformation, and take T0 = I, B0 = A0 . Blocking Tn = [Tn ], and

similarly An , Bn , according to the block structure of A0 , the equations

(2.18) for n ≥ 1 are of the form

(jk) (jk) (jk) (jk)

Tn (Akk + nI) ’ Ajj Tn = An ’ Bn + . . .

According to Lemma 24 (p. 212), whenever Akk + nI and Ajj do not have

(jk)

the same eigenvalue, these equations can be solved for Tn for whatever

(jk)

block Bn we have, and we then choose such blocks equal to vanish. In

the other case we apply Lemma 25 (p. 213) to see that the equation be-

(jk)

comes solvable if we pick Bn accordingly. This second case occurs only

for ¬nitely many values of n and for some j > k, and one can see that the

transformed system so obtained has a fundamental solution z K z M , with K

3

and M as described in the theorem.

2.4 Systems with General Spectrum 33

A de¬nition frequently used in the literature is as follows: Given a system

having an isolated singularity at some point z0 , we say that z0 is a regular-

singular point, if a fundamental solution X(z) = S(z) (z ’ z0 )M exists

whose single-valued part S(z) has at most a pole at z0 , or in other words:

if X(z) cannot grow faster than some negative power of |z ’z0 | as z ’ z0

in, e.g., | arg(z’z0 )| ¤ π. Note that then the same statements hold for other

fundamental solutions as well. Using this terminology, we may summarize

the results obtained in this chapter as follows:

1. Every singularity of ¬rst kind is a regular-singular point.

2. In case of good spectrum, there is a unique fundamental solution of

(2.1) of the form (2.2), and the coe¬cients Sn can be recursively

computed from (2.3).

3. In case of general spectrum, there exists a fundamental solution of

(2.1) of the form (2.17), and the matrices M and K and the coe¬-

cients Tn can be computed as described in Remark 1.

So we have given quite satisfactory solutions to the problems stated at the

beginning of this chapter. It is worthwhile to emphasize that the converse

of statement 1 does not hold, as follows from Exercise 1. To ¬nd an e¬ective

algorithmic procedure for checking whether a system has a regular-singular

point at z0 is not a trivial matter and has attracted the attention of re-

searchers for quite some time. In principle, every procedure ¬nding the

so-called formal fundamental solution that we shall de¬ne later can also

serve as a way of determining the nature of the singularity. So many pa-

pers concerned with e¬ective calculation of formal solutions fall into this

category as well. From the more recent work in this direction, we mention,

e.g, Turrittin [268], Moser [195], Lutz [171“173], Deligne[84], Jurkat and

Lutz [145], Harris [114], Wagenf¨hrer [276, 277], and Dietrich [85“87].

u

In particular, the computer algebra packages mentioned in Section 13.5

may be used very e¬ectively to check whether a singularity of positive

Poincar´ rank is regular-singular. For scalar equations, however, we shall

e

obtain a very easy criterion for regular-singular points. So in this context,

scalar equations are much “better behaved” than systems.

Exercises:

1. For an arbitrary constant M and a diagonal matrix K of integer

diagonal entries, let A(z) = K + z K M z ’K . Show that the origin is

a regular-singular point of zx = A(z) x, but in general a singularity

of the second kind.

2. Let a con¬‚uent hypergeometric system (2.5) in dimension ν = 2, with

B = diag [0, 1] and A as in (2.7), be given. Use the procedure outlined

in Remark 1 to ¬nd explicitly the matrices M and K, and recursion

34 2. Singularities of First Kind

equations for the entries of the matrices Tn , for a fundamental solu-

tion of the form (2.17).

2.5 Scalar Higher-Order Equations

Consider a νth-order linear ODE (1.6) (p. 4), with coe¬cients ak (z) holo-

morphic in some punctured disc R(0, ρ) about the origin. As for systems,

we say that the origin is a regular-singular point of (1.6), if all solutions do

not grow faster than some inverse powers of |z|, as z ’ 0 in | arg z| ¤ π.

Moreover, we call the origin a singularity of the ¬rst kind for (1.6), if ak (z)

has at most a pole of kth order there, for 1 ¤ k ¤ ν. To see how this

de¬nition relates to the system case, see Exercise 1.

In contrast to the case of systems, for scalar equations regular-singular

points and singularities of the ¬rst kind are the same:

Theorem 7 The origin is a regular-singular point for (1.6) if and only if

it is a singularity of the ¬rst kind.