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nв€’1
(21)
+A(12) в€’ TВµ Anв€’Вµв€’m Tm ,
n
Вµ=1 m=1

for n в‰Ґ 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): Deп¬Ѓning 3
nв€’Вµ
nв€’1
в€’Вµ
K в€’m tm n в‰Ґ 1,
n
tn = c K 1+ K tВµ 2 + ,
Вµ=1 m=1

for K > 1/ПЃ and suп¬ѓciently large c > 0, we show by induction Tn в‰¤
в€ћ
tn , n в‰Ґ 1. Putting f (z) = n
1 tn (z/K) , one can formally obtain the
quadratic equation (1 в€’ z) f (z) = zc [1 + 2f (z)] + c f 2 (z). Its solutions are

fВ± (z) = (2c)в€’1 1 в€’ z(2c + 1) В± [1 в€’ z(2c + 1)]2 в€’ 4c2 z ,

which both are holomorphic near the origin. Moreover, fв€’ (0) = 0, and
the coeп¬ѓcients of its power series expansion satisfy the same recursion as,
so are in fact equal to, the numbers tn K в€’n . This proves convergence of
в€ћ
1 tn z , for suп¬ѓciently 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 coeп¬ѓcient
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 в‰¤ tn , and this majorizing sequence would clearly give a better estimate. However,
then the corresponding function f (z) is a solution of a Riccati diп¬Ђerential equation,
instead of a simple quadratic equation. Hence proving analyticity is more complicated.
30 2. Singularities of First Kind

B11 (z) = A11 (z) в€’ 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) в‰Ў 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 п¬Ѓnd 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
(ОЅ1 , . . . , ОЅВµ ), with Ajj (0) having exactly one eigenvalue О»j , and assume that
these eigenvalues have increasing real parts. Moreover, let Akj (0) = 0 for
k > j. Finally, let Оєj в€€ Z be such that 0 в‰¤ Re О»j в€’ Оєj < 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) в€’ Оєj I, and K = diag [Оє1 IОЅ1 , . . . , ОєВµ IОЅВµ ].
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) в€’ Оєj I; hence, according to the
uniqueness part of Theorem 71, we have Mjj = Ajj (0) в€’ Оєj I. Deп¬Ѓning
T (z) = X(z) z в€’M z в€’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 deп¬Ѓnition of T (z) implies

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

with B(z) = K + z K M z в€’K . Since Оє1 , . . . , ОєВµ 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 в‰¤ j < k в‰¤ ОЅ, we obtain from (2.15) that

z Tkj (z) = Akk (z) Tkj (z) в€’ 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 в€’ j = 1, since then Rkj (z) involves only
diagonal blocks of T (z). Then, let m в‰Ґ 0 be the pole order of Tkj (z) and
(kj)
Tв€’m denote the corresponding coeп¬ѓcient in its Laurent expansion. If m
(kj) (kj)
were positive, then (2.16) would imply [mI + Akk (0)] Tв€’m = Tв€’m Ajj (0).
(kj)
This, however, would imply Tв€’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 в€’ j we
2
п¬Ѓnd 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 п¬Ѓnd 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 eп¬ѓcient 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

вЂў T (z) is an analytic transformation,

вЂў M is constant, lower triangularly blocked of some type (ОЅ1 , . . . , ОЅВµ ),

вЂў the kth diagonal block of M has exactly one eigenvalue mk with real
part in the half-open interval [0, 1),

вЂў K = diag [Оє1 IОЅ1 , . . . , ОєВµ IОЅВµ ], Оєj в€€ Z, weakly increasing and so that
mk + Оєk is an eigenvalue of A0 with algebraic multiplicity ОЅk .

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 clariп¬Ѓes 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 п¬Ѓnite number of coeп¬ѓcients of T (z).
For a more eп¬Ђective computation, one may use the following procedure
from Gantmacher , which also provides another proof of Theorem 6:
в€ћ
Remark 1: Let (2.1) and an analytic transformation T (z) = 0 Tn z n be
в€ћ
arbitrarily given, and expand B(z) = 0 Bn z n . Then (2.13) is equivalent
to
nв€’1
Tn (B0 + nI) в€’ A0 Tn = (Anв€’m Tm в€’ Tm Bnв€’m ), (2.18)
m=0

for n в‰Ґ 0. We assume that A0 = diag [A11 , . . . , AВµВµ ] 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 в‰Ґ 1 are of the form
(jk) (jk) (jk) (jk)
Tn (Akk + nI) в€’ Ajj Tn = An в€’ 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 п¬Ѓnitely 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 deп¬Ѓnition 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 в€’ 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 в€’z0 | as z в†’ z0
in, e.g., | arg(zв€’z0 )| в‰¤ ПЂ. 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 п¬Ѓrst 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 coeп¬ѓcients 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 coeп¬ѓ-
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 п¬Ѓnd an eп¬Ђective
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 п¬Ѓnding the
so-called formal fundamental solution that we shall deп¬Ѓne later can also
serve as a way of determining the nature of the singularity. So many pa-
pers concerned with eп¬Ђective calculation of formal solutions fall into this
category as well. From the more recent work in this direction, we mention,
e.g, Turrittin , Moser , Lutz [171вЂ“173], Deligne, Jurkat and
Lutz , Harris , WagenfВЁhrer [276, 277], and Dietrich [85вЂ“87].
u
In particular, the computer algebra packages mentioned in Section 13.5
may be used very eп¬Ђectively to check whether a singularity of positive
PoincarВґ 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 вЂњbetter behavedвЂќ 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 в€’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 conп¬‚uent hypergeometric system (2.5) in dimension ОЅ = 2, with
B = diag [0, 1] and A as in (2.7), be given. Use the procedure outlined
in Remark 1 to п¬Ѓnd 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 ОЅth-order linear ODE (1.6) (p. 4), with coeп¬ѓcients ak (z) holo-
morphic in some punctured disc R(0, ПЃ) 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 в†’ 0 in | arg z| в‰¤ ПЂ.
Moreover, we call the origin a singularity of the п¬Ѓrst kind for (1.6), if ak (z)
has at most a pole of kth order there, for 1 в‰¤ k в‰¤ ОЅ. To see how this
deп¬Ѓnition 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 п¬Ѓrst 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 п¬Ѓrst kind.

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