propriate sizes, and

j’1

Yjk (u) = Aj„ (u) X„ k (u).

„ =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: Di¬erentiation of (1.15) and insertion into (1.1) proves that the

˜

above X(z) is a fundamental solution of (1.1). If X(z) is any lower tri-

˜

angularly blocked fundamental solution of (1.1), let Xkk (z) = Xkk (z) and

’1 ˜ ˜

Cjk = Xjj (z0 ) Xjk (z0 ), 1 ¤ k < j ¤ µ. Then X(z0 ) = X(z0 ), hence

˜

X(z) ≡ X(z). 2

Observe that in case of νk = 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 νk 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 modi¬cations, gen-

eralizes to upper triangularly blocked systems.

2. For ν = 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 ¬nitely many

integrations.

3. For G = R(0, ρ), ρ > 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 ¤ k ¤ µ. For X(z) as in the above theorem, show that

we can explicitly ¬nd a lower triangularly blocked monodromy factor

in terms of the integration constants Cjk and ¬nitely many de¬nite

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 ¬nding a funda-

mental solution of a system of dimension ν ’ µ 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 ∈ G, exists such that

B (z) = A(z) and A(z) B(z) = B(z) A(z) hold for every z ∈ 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 coe¬cients, or systems with

diagonal coe¬cient 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 “ 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 coe¬cient matrix A(z) has a pole of order

r + 1 at z0 , and the non-negative integer r then is named the Poincar´ 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

coe¬cient 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 su¬ces to take G equal to

a punctured disc R(z0 , ρ), for some ρ > 0. We shall see that the cases of

Poincar´ rank r = 0, i.e., a simple pole of A(z) at z0 , are essentially di¬erent

e

from r ≥ 1, and we follow the standard terminology in referring to the ¬rst

resp. second case by saying that (1.1) has a singularity of ¬rst resp. of second

kind at z0 . If A(z) is holomorphic in R(∞, ρ) = {z : |z| > ρ}, we say, in

view of 3 Exercise 2 that (1.1) has rank r at in¬nity if B(z) = ’z ’2 A(1/z)

has rank r at the origin. Accordingly, in¬nity is a singularity of ¬rst, resp.

second, kind of (1.1), if zA(z) is holomorphic, resp. has a pole, at in¬nity.

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 , ρ), for some ρ > 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 ’ z0 ) over G.

1. For B(z) = A(z + z0 ), z ∈ R(0, ρ), 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) = ’z ’2 A(1/z), z ∈ R(∞, 1/ρ) = {z : |z| >

1/ρ}, 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 ’ bc = 0)

w = w(z) =

cz + d

˜

be a M¨bius transformation, and take G as the preimage of G under

o

the (bijective) mapping z ’ w(z) of the compacti¬ed complex plane

˜

C ∪ {∞}. For simplicity, assume a/c ∈ G, to ensure ∞ ∈ G. For

arbitrary A(z), holomorphic in G, de¬ne

ad ’ bc ˜

z ∈ 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(∞, ρ), let A(z) have Poincar´ rank r ≥ 1 at in¬nity, and

e

1’r

set ra = sup|z|≥ρ+µ z A(z) , for some µ > 0. For

¯

S = {z : |z| ≥ ρ + µ, ± ¤ arg z ¤ β },

with arbitrary ± < β, show that for every fundamental solution X(z)

of (1.1) one can ¬nd c > 0 so that

r

¯

X(z) ¤ c ea|z| , z ∈ S.

5. For every dimension ν ≥ 1, ¬nd an elementary system of Poincar´ e

rank r ≥ 1 at in¬nity for which the estimate in the previous exercise

is sharp.

6. For every dimension ν ≥ 2, ¬nd an elementary system of Poincar´ e

rank r ≥ 1 at in¬nity for which fundamental solutions only grow like

a power of z; hence the estimate in Exercise 4 is not sharp. Check

that for ν = 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 ν ≥ 1, ¬nd a system (1.1) that is meromorphic

in some region G, with in¬nitely 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 ¬rst kind, i.e., a pole of ¬rst 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), ¬nd a monodromy ma-

trix at z0 ; i.e., ¬nd M so that X(z) = S(z) (z ’ z0 )M , with S(z)

holomorphic and single-valued in 0 < |z ’ z0 | < ρ, for some ρ > 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 coe¬cients in the Laurent expansion of S(z) about the point

z0 , or more precisely, ¬nd equations that allow the computation of at

least ¬nitely many such coe¬cients.

The following observations are very useful in order to simplify the investi-

gations we have in mind:

• Suppose that X(z) = S(z) (z ’ z0 )M is some fundamental solution of

(1.1), then according to Exercise 1 on p. 7 every other fundamental

˜

˜ ˜ ˜

solution is obtained as X(z) = X(z) C = S(z) (z ’ z0 )M , with S(z) =

S(z) C, M = C ’1 M C, for a unique invertible matrix C. Therefore we

˜

conclude that it su¬ces to solve the above problems for one particular

fundamental solution X(z).

18 2. Singularities of First Kind

• We shall see that for singularities of ¬rst 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 ’ 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 di¬erence 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.

• According to the exercises in Section 1.6, we may without loss in gen-

erality make z0 equal to any preassigned point in the compacti¬ed

complex plane C ∪ {∞}. For singularities of ¬rst kind it is customary

to choose z0 = 0, and we shall follow this convention. Moreover, we

also adopt the custom to consider the di¬erential 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

∞

|z| < ρ.

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, ρ), with ρ > 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 di¬er

by a natural number, or in other words, if A0 + nI and A0 , for every

n ∈ 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 o¬ its behavior at the origin.

For the other cases we shall obtain a similar, but more complicated result.

The theory of singularities of ¬rst 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 [84] and Yoshida [290]. 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 Erd´lyi [100], Sch¨fke [235], Magnus, Oberhettinger,

e a

and Soni [180], and Iwasaki, Kimura, Shimomura, and Yoshida [141].

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

∞

|z| < ρ.

A0

Sn z n , S0 = I,

X(z) = S(z) z , S(z) = (2.2)

n=0

The coe¬cients Sn are uniquely determined by the relations

n’1

Sn (A0 + nI) ’ A0 Sn = n ≥ 1.

An’m Sm , (2.3)

m=0

Proof: Inserting the “Ansatz” (2.2) into (2.1) and comparing coe¬cients

easily leads to the recursion equations (2.3). Lemma 24 (p. 212) implies

that the coe¬cients Sn are uniquely determined by (2.3), owing to as-

sumption E. Hence we are left to show that the resulting power series for