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with arbitrarily chosen z0 в€€ G, arbitrary constant matrices Cjk of the ap-
propriate sizes, and
jв€’1
Yjk (u) = AjП„ (u) XП„ k (u).
П„ =k

Conversely, every lower triangularly blocked fundamental solution of (1.1)
is obtained by (1.15), if the Xkk (z) and the constants of integration Cjk are
appropriately selected.

Proof: Diп¬Ђerentiation of (1.15) and insertion into (1.1) proves that the
Лњ
above X(z) is a fundamental solution of (1.1). If X(z) is any lower tri-
Лњ
angularly blocked fundamental solution of (1.1), let Xkk (z) = Xkk (z) and
в€’1 Лњ Лњ
Cjk = Xjj (z0 ) Xjk (z0 ), 1 в‰¤ k < j в‰¤ Вµ. Then X(z0 ) = X(z0 ), hence
Лњ
X(z) в‰Ў X(z). 2
Observe that in case of ОЅk = 1 for every k one can easily compute the
diagonal blocks Xkk (z) by solving the corresponding scalar ODE. Hence
triangular systems can always be solved. If one or several ОЅk are at least
two, this is no longer so, but at least we can say that in a clear sense
reduced systems are easier to solve than general ones.

Exercises:
1. Verify that the above theorem, with some obvious modiп¬Ѓcations, gen-
eralizes to upper triangularly blocked systems.
2. For ОЅ = 2 and
a(z) 0
A(z) = ,
b(z) c(z)
with a(z), b(z), c(z) holomorphic in a simply connected region G, ex-
plicitly compute a fundamental solution of (1.1), up to п¬Ѓnitely many
integrations.
3. For G = R(0, ПЃ), ПЃ > 0, let A(z) as in (1.13) be holomorphic in G,
and let Xkk (z) be fundamental solutions of (1.14) with monodromy
factors Ck , 1 в‰¤ k в‰¤ Вµ. For X(z) as in the above theorem, show that
we can explicitly п¬Ѓnd a lower triangularly blocked monodromy factor
in terms of the integration constants Cjk and п¬Ѓnitely many deп¬Ѓnite
integrals.
14 1. Basic Properties of Solutions

4. Under the assumptions of Exercise 8 on p. 7, show that a computation
of a fundamental solution of (1.1) is equivalent to п¬Ѓnding a funda-
mental solution of a system of dimension ОЅ в€’ Вµ and an additional
integration.

1.6 Some Additional Notation
It will be convenient for later use to say that a system (1.1) is elementary
if a matrix-valued holomorphic function B(z), for z в€€ G, exists such that
B (z) = A(z) and A(z) B(z) = B(z) A(z) hold for every z в€€ G. As was
shown in Exercise 3 on p. 7 and the following one, an elementary system
has the fundamental solution X(z) = exp[B(z)]; however, such a B(z)
does not always exist, so not every system is elementary. Simple examples
of elementary systems are those with constant coeп¬ѓcients, or systems with
diagonal coeп¬ѓcient matrix, or the ones studied in Exercise 2 on p. 7.
In the last century or so, one of the main themes of research has been on
the behavior of solutions of (1.1) near an isolated boundary point z0 of the
region G вЂ“ assuming that such points exist, which implies that G will be
multiply connected. In particular, many classical as well as recent results
concern the situation where the coeп¬ѓcient matrix A(z) has a pole of order
r + 1 at z0 , and the non-negative integer r then is named the PoincarВґ rank
e
of the system. Relatively little work has been done in case of an essential
singularity of A(z) at z0 , and we shall not consider such cases here at all.
It is standard to call a system (1.1) a meromorphic system on G if the
coeп¬ѓcient matrix A(z) is a meromorphic function on G; i.e., every point of
G is either a point of holomorphy or a pole of A(z).
In the following chapters we shall study the local behavior of solutions
of (1.1) near a pole of A(z), and to do so it suп¬ѓces to take G equal to
a punctured disc R(z0 , ПЃ), for some ПЃ > 0. We shall see that the cases of
PoincarВґ rank r = 0, i.e., a simple pole of A(z) at z0 , are essentially diп¬Ђerent
e
from r в‰Ґ 1, and we follow the standard terminology in referring to the п¬Ѓrst
resp. second case by saying that (1.1) has a singularity of п¬Ѓrst resp. of second
kind at z0 . If A(z) is holomorphic in R(в€ћ, ПЃ) = {z : |z| > ПЃ}, we say, in
view of 3 Exercise 2 that (1.1) has rank r at inп¬Ѓnity if B(z) = в€’z в€’2 A(1/z)
has rank r at the origin. Accordingly, inп¬Ѓnity is a singularity of п¬Ѓrst, resp.
second, kind of (1.1), if zA(z) is holomorphic, resp. has a pole, at inп¬Ѓnity.
Observe that the same holds when classifying the nature of singularity at
the origin! This fact is one of the reasons that, instead of systems (1.1), we
shall from now on consider systems obtained by multiplying both sides of
(1.1) by z.

3 Observe
that references to exercises within the same section are made by just giving
their number.
1.6 Some Additional Notation 15

Exercises: In what follows, let G = R(z0 , ПЃ), for some ПЃ > 0, and let A(z)
be holomorphic in G with at most a pole at z0 . Recall from Section 1.3 that
solutions of (1.1) in this case are holomorphic functions on the Riemann
surface of log(z в€’ z0 ) over G.

1. For B(z) = A(z + z0 ), z в€€ R(0, ПЃ), and vector functions x(z), y(z)
connected by y(z) = x(z + z0 ), show that x(z) is a solution of (1.1)
if and only if y(z) solves y = B(z) y.

2. For z0 = 0, B(z) = в€’z в€’2 A(1/z), z в€€ R(в€ћ, 1/ПЃ) = {z : |z| >
1/ПЃ}, and x(z), y(z) connected by y(z) = x(1/z), show that x(z) is a
solution of (1.1) if and only if y(z) solves y = B(z) y.

3. More generally, let G be an arbitrary region, let

az + b
(ad в€’ bc = 0)
w = w(z) =
cz + d
Лњ
be a MВЁbius transformation, and take G as the preimage of G under
o
the (bijective) mapping z в†’ w(z) of the compactiп¬Ѓed complex plane
Лњ
C в€Є {в€ћ}. For simplicity, assume a/c в€€ G, to ensure в€ћ в€€ G. For
arbitrary A(z), holomorphic in G, deп¬Ѓne

ad в€’ bc Лњ
z в€€ G.
B(z) = A(w(z)), y(z) = x(w(z)),
(cz + d)2

Show that x(z) is a solution of (1.1) if and only if y(z) solves y =
B(z) y.

4. Let G = R(в€ћ, ПЃ), let A(z) have PoincarВґ rank r в‰Ґ 1 at inп¬Ѓnity, and
e
1в€’r
set ra = sup|z|в‰ҐПЃ+Оµ z A(z) , for some Оµ > 0. For

ВЇ
S = {z : |z| в‰Ґ ПЃ + Оµ, О± в‰¤ arg z в‰¤ ОІ },

with arbitrary О± < ОІ, show that for every fundamental solution X(z)
of (1.1) one can п¬Ѓnd c > 0 so that
r
ВЇ
X(z) в‰¤ c ea|z| , z в€€ S.

5. For every dimension ОЅ в‰Ґ 1, п¬Ѓnd an elementary system of PoincarВґ e
rank r в‰Ґ 1 at inп¬Ѓnity for which the estimate in the previous exercise
is sharp.

6. For every dimension ОЅ в‰Ґ 2, п¬Ѓnd an elementary system of PoincarВґ e
rank r в‰Ґ 1 at inп¬Ѓnity for which fundamental solutions only grow like
a power of z; hence the estimate in Exercise 4 is not sharp. Check
that for ОЅ = 1 the estimate always is sharp.
16 1. Basic Properties of Solutions

7. Consider a system (1.1) that is meromorphic in G, and let z0 be a
pole of A(z). If it so happens that a fundamental solution exists which
only has a removable singularity at z0 , we say that z0 is an apparent
singularity of (1.1). Check that then every fundamental solution X(z)
must be holomorphic at z0 , but det X(z0 ) = 0.
8. For every dimension ОЅ в‰Ґ 1, п¬Ѓnd a system (1.1) that is meromorphic
in some region G, with inп¬Ѓnitely many apparent singularities in G.
2
Singularities of First Kind

Throughout this chapter, we shall be concerned with a system (1.1) (p. 2)
having a singularity of п¬Ѓrst kind, i.e., a pole of п¬Ѓrst order, at some point z0 ,
and we wish to study the behavior of solutions near this point. In particular,
we wish to solve the following problems as explicitly as we possibly can:

P1) Given a fundamental solution X(z) of (1.1), п¬Ѓnd a monodromy ma-
trix at z0 ; i.e., п¬Ѓnd M so that X(z) = S(z) (z в€’ z0 )M , with S(z)
holomorphic and single-valued in 0 < |z в€’ z0 | < ПЃ, for some ПЃ > 0.

P2) Determine the kind of singularity that S(z) has at z0 ; i.e., decide
whether this singularity is removable, or a pole, or an essential one.

P3) Find the coeп¬ѓcients in the Laurent expansion of S(z) about the point
z0 , or more precisely, п¬Ѓnd equations that allow the computation of at
least п¬Ѓnitely many such coeп¬ѓcients.

The following observations are very useful in order to simplify the investi-
gations we have in mind:

вЂў Suppose that X(z) = S(z) (z в€’ z0 )M is some fundamental solution of
(1.1), then according to Exercise 1 on p. 7 every other fundamental
Лњ
Лњ Лњ Лњ
solution is obtained as X(z) = X(z) C = S(z) (z в€’ z0 )M , with S(z) =
S(z) C, M = C в€’1 M C, for a unique invertible matrix C. Therefore we
Лњ
conclude that it suп¬ѓces to solve the above problems for one particular
fundamental solution X(z).
18 2. Singularities of First Kind

вЂў We shall see that for singularities of п¬Ѓrst kind the matrix S(z) never
has an essential singularity at z0 . Whether it has a pole or a remov-
able one depends on the selection of the monodromy matrix, since
instead of M we can also choose M в€’ kI, for every integer k, and
accordingly replace S(z) by z k S(z). Hence in a way, poles and re-
movable singularities of S(z) should not really be distinguished in
this context. It will, however, make a diп¬Ђerence whether or not S(z)
has a removable singularity and at the same time det S(z) does not
vanish at z0 , meaning that the power series expansion of S(z) begins
with an invertible constant term.
вЂў According to the exercises in Section 1.6, we may without loss in gen-
erality make z0 equal to any preassigned point in the compactiп¬Ѓed
complex plane C в€Є {в€ћ}. For singularities of п¬Ѓrst kind it is customary
to choose z0 = 0, and we shall follow this convention. Moreover, we
also adopt the custom to consider the diп¬Ђerential operator z (d/dz)
instead of just the derivative d/dz; this has advantages, e.g., when
making a change of variable z = 1/u (see Section 1.6). As a conse-
quence, we shall here consider a system of the form
в€ћ
|z| < ПЃ.
An z n ,
z x = A(z) x, A(z) = (2.1)
n=0

Hence in other words, A(z) is a holomorphic matrix function in
D(0, ПЃ), with ПЃ > 0, and we shall have in mind that A0 = 0, although
all results remain correct for A0 = 0 as well.

As we shall see, the following condition upon the spectrum of A0 will be
very important:

E) We say that (2.1) has good spectrum if no two eigenvalues of A0 diп¬Ђer
by a natural number, or in other words, if A0 + nI and A0 , for every
n в€€ N, have disjoint spectra. Observe that we do not regard 0 as a
natural number; thus it may be that A0 has equal eigenvalues!

For systems with good spectrum we shall see that A0 will be a monodromy
matrix for some fundamental solution X(z), and we shall obtain a repre-
sentation for X(z) from which we can read oп¬Ђ its behavior at the origin.
For the other cases we shall obtain a similar, but more complicated result.
The theory of singularities of п¬Ѓrst kind is covered in most books dealing
with ODE in the complex plane. In addition to those mentioned in Chap-
ter 1, one can also consult Deligne  and Yoshida . In this chapter, we
shall also introduce some of the special functions which have been studied
in the past. For more details, and other functions which are not mentioned
here, see the books by ErdВґlyi , SchВЁfke , Magnus, Oberhettinger,
e a
and Soni , and Iwasaki, Kimura, Shimomura, and Yoshida .
2.1 Systems with Good Spectrum 19

2.1 Systems with Good Spectrum
Here we prove a well-known theorem saying that for systems with good
spectrum the matrix A0 always is a monodromy matrix. Moreover, a fun-
damental solution can in principle be computed in a form from which the
behavior of solutions near the origin may be deduced:

Theorem 5 Every system (2.1) with good spectrum has a unique funda-
mental solution of the form
в€ћ
|z| < ПЃ.
A0
Sn z n , S0 = I,
X(z) = S(z) z , S(z) = (2.2)
n=0

The coeп¬ѓcients Sn are uniquely determined by the relations
nв€’1
Sn (A0 + nI) в€’ A0 Sn = n в‰Ґ 1.
Anв€’m Sm , (2.3)
m=0

Proof: Inserting the вЂњAnsatzвЂќ (2.2) into (2.1) and comparing coeп¬ѓcients
easily leads to the recursion equations (2.3). Lemma 24 (p. 212) implies
that the coeп¬ѓcients Sn are uniquely determined by (2.3), owing to as-
sumption E. Hence we are left to show that the resulting power series for
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