origin. Then the equivalent system constructed in Exercise 1 has a singu-

larity of the ¬rst kind as well, hence a regular-singular point according to

Theorem 6. This in turn implies a regular-singular point for (1.6).

Conversely, assume that (1.6) has a regular-singular point at the origin.

From Exercise 9 on p. 10, we conclude existence of a solution of the form

y(z) = s(z) z µ , with µ ∈ C and s(z) single-valued at the origin. Owing to

the assumption of a regular-singular point, s(z) cannot have an essential

singularity at the origin. Rede¬ning µ mod 1, we therefore may assume

∞

|z| < ρ,

yn z n+µ ,

y(z) =

n=0

with y0 = 0. As for systems, we call such a solution y(z) a Floquet solution

of (1.6). Making ρ smaller if necessary, we may also assume that y(z) does

not vanish on R(0, ρ). We now complete the proof by induction with respect

to ν: For ν = 1, we obtain from (1.6) that z a1 (z) = zy (z)/y(z), and on the

right one can cancel z µ . Thus z a1 (z) is the quotient of two holomorphic

functions, with a nonzero denominator, hence holomorphic at the origin.

So for ν = 1 we completed the proof. For larger ν, we substitute y = y(z)˜ y

into (1.6), use that y(z) is a solution to cancel two terms, and then divide

by y(z), to obtain (setting a0 (z) ≡ ’1):

ν’1 ν

k y (k’j) (z)

(ν) (j)

y

˜ = y

˜ aν’k (z).

j y(z)

j=1 k=j

2.5 Scalar Higher-Order Equations 35

This is an ODE for y , and its order is ν ’1. Observe that the factor z µ can

˜

(k’j)

be canceled from y (z)/y(z), which then becomes single-valued at the

origin, with a pole of order j ’k. So the coe¬cients of this new equation are

holomorphic in R(0, ρ). By assumption, no solution y (hence: no y , or its

˜

derivative) can grow faster than powers of z, so by induction hypothesis,

the new ODE has a singularity of the ¬rst kind at the origin. Thus we

conclude that the coe¬cient of y (j) can at most have a pole of order ν ’ j

˜

at the origin. This information can be used to conclude by induction that

each aν’k (z) can at most have a pole of order ν ’ k. 2

According to Exercise 3, one can always recursively compute the coe¬-

cients for at least one Floquet solution of (1.6). The proof of the previous

theorem then shows that other solutions can, in principle, be obtained

through an equation of order ν ’ 1. This, however, is not a very e¬cient

procedure.

The Floquet exponents µ are roots of the so-called indicial equation in-

troduced in Exercise 2. A fundamental solution of (1.6), aside from Floquet

solutions, consists of so-called logarithmic solutions of the form

j

µ

sk (z) [log z]k ,

y(z) = z

k=0

where µ is one of the Floquet exponents, j is strictly smaller than its

multiplicity, and the sk (z) are holomorphic at the origin. An elegant method

to compute ¬nitely many power series coe¬cients of these functions goes

under the name Frobenius™ method. In the exercises below we have described

an essential part of this method; for the cases not included there, see [82].

In any case, for ν ≥ 3, one will only in exceptional cases succeed in ¬nding

all power series coe¬cients of the functions sk (z) in closed form.

Exercises: For the following exercises, let a νth-order linear ODE (1.6)

in G = R(0, ρ), ρ > 0, be given.

1. De¬ning x = [y, zy , . . . , z ν’1 y (ν’1) ]T , and

®

0 1 0 ... 0 0

0

1 1 ... 0 0

0

0 2 ... 0 0

A(z) = . ,

. . . .

..

.

. . . .

.

. . . . .

°0 »

... ν ’ 2

0 0 1

bν (z) bν’1 (z) bν’2 (z) . . . b2 (z) b1 (z) + ν ’ 1

with bk (z) = z k ak (z), show that y(z) solves (1.6) if and only if the

corresponding x(z) is a solution of (2.1) (in particular, show that a

solution vector x(z) of (2.1) always is of the above form).

36 2. Singularities of First Kind

2. Assume (1.6) to have a singularity of the ¬rst kind at the origin, i.e.,

b(k) z n , |z| < ρ, 1 ¤ k ¤ ν.

bk (z) = z k ak (z) = n

n≥0

Let y(z) = n yn z n+µ , y0 = 0, µ ∈ C , be a solution of (1.6). Show

that then the Floquet exponent µ satis¬es the indicial equation

ν

(k)

[µ]ν = b0 [µ]ν’k ,

k=1

with [µ]0 = 1, [µ]k = µ(µ ’ 1) · . . . · (µ ’ k + 1), for k ∈ N.

3. For bk (z) and y(z) as above, ¬nd the equations that the coe¬cients

yn have to satisfy so that y(z) is a solution of (1.6). Show that for

at least one solution µ of the indicial equation one can recursively

compute the coe¬cients from these equations. In this way, we obtain

a constructive proof for existence of at least one Floquet solution.

4. For bk (z) as above, let w be a complex variable.

(k)

ν

(a) With p0 (w) = [w]ν ’ k=1 b0 [w]k , show that the inhomoge-

neous equation

ν

ν (ν)

z ν’k bk (z) y (ν’k) = p0 (w) z w

’

zy

k=1

∞

has a unique solution y(z; w) = 0 yn (w) z n+w , with y0 (w) ≡ 1

and coe¬cients yn (w), n ≥ 1, which are rational functions of w.

Find the possible poles of pn (w) in terms of the roots of p0 (w).

(b) Let µ be a root of p0 (w), but so that p0 (µ + j) = 0 for j ∈ N.

Conclude that then y(z; µ) is a Floquet solution of (1.6). Show

that for at least one root of p0 (w) this assumption holds.

(c) Let µ, as above, be a root of multiplicity at least k ≥ 2. Show

that then (1.6) has a solution of the form

∞

k’1

k’1 (j)

|z| < ρ.

[log z]k’1’j yn (µ) z n+µ ,

j n=0

j=0

Show that all solutions so obtained are linearly independent.

5. For the hypergeometric, resp. Bessel™s, equation, ¬nd the cases where

the indicial equation has a double root. Use the previous exercise to

compute a fundamental solution for these cases.

3

Highest-Level Formal Solutions

In this and later chapters, we are concerned with systems having a singu-

larity of the second kind at some point z0 . As in the case of singularities

of the ¬rst kind, we may without loss of generality assume that z0 is any

preassigned point in C ∪ {∞}. Both for historical reasons as well as nota-

tional convenience it is customary to choose z0 = ∞ here. Hence we shall

be dealing with systems of the form

∞

An z ’n , |z| > ρ,

r

z x = A(z) x, A(z) = z (3.1)

n=0

with Poincar´ rank r ≥ 1. Observe that we usually assume the leading term

e

A0 to be nonzero, but sometimes we may apply certain transformations,

producing a new system with vanishing leading term. In this case we may

say that we have lowered the Poincar´ rank of the system.

e

In principle, we should like to solve the same three problems posed at

the beginning of the previous chapter, but we shall see that things are

considerably more complicated here: We know that fundamental solutions

of (3.1) are of the form X(z) = S(z) z M , with a single-valued matrix S(z)

that in general will have an essential singularity at in¬nity. Expanding

∞

S(z) = ’∞ Sn z ’n , inserting into (3.1), and comparing coe¬cients gives

∞

Sn’r (M ’ (n ’ r)I) = n ∈ Z.

Am Sn’m ,

m=0

This is a homogeneous system of in¬nitely many equations in in¬nitely

many unknowns; and the also unknown matrix M may be regarded as a

38 3. Highest-Level Formal Solutions

matrix eigenvalue for this system. In the case of a singularity of the ¬rst

kind we have seen that S(z) cannot have an essential singular point at

in¬nity; hence Sn = 0 for small n ∈ Z. For singularities of the second

kind, however, this is no longer true. Therefore, although we know that for

some M this system has a solution (Sn )∞ for which S(z) converges and

’∞

det S(z) = 0 for su¬ciently large |z|, to ¬nd M and (Sn ) is much more

di¬cult. There is a theory of in¬nite determinants (see von Koch [153])

that might be applied here. We shall not do this, however, because even

if we had computed M and the coe¬cients Sn , we would not obtain any

detailed information on the behavior of solutions for z ’ ∞. Instead, we

will show existence of certain transformations that will block-diagonalize

the system (3.1). Hence in principle we will reduce the task of studying the

nature of solutions of (3.1) near the point in¬nity to the same problem for

simpler, i.e., smaller, systems. However, some of the transformations will

be formal in the sense that they look like analytic transformations, but the

radius of convergence of the power series will in general be equal to zero.

So at ¬rst glance, the usefulness of these transformations is questionable,

but in several later chapters we shall show that they can nonetheless be

given a clear meaning.

Many of the books listed in Chapter 1 also cover the theory of singu-

larities of the second kind. In addition, we mention the survey articles

by Brjuno [74], Kimura [151], Malgrange [182, 183], Hukuhara [131, 132],

Bertrand [51], and Varadarajan [274, 275]. For presentations using more

algebraic tools, see Gerard and Levelt [106], Levelt [167], or Babbitt and

Varadarajan [3].

3.1 Formal Transformations

Since we are now working in a neighborhood of in¬nity, we have to adjust

the notion of analytic transformations accordingly. Moreover, we shall have

reason to use other types of transformations as well, and we start by listing

such transformations x = T (z)˜ employed in this chapter:

x

∞

1. If T (z) = 0 Tn z ’n has positive radius of convergence and det T0 =

0, then we shall call T (z), or more precisely the change of the depen-

dent variable x = T (z)˜, an analytic transformation (at the point

x

in¬nity). This coincides with the de¬nition in Section 2.4, up to the

change of variable z ’ 1/z. If only ¬nitely many coe¬cients Tn are

di¬erent from zero, we say that the transformation terminates.

∞

2. If T (z) = 0 Tn z ’n , with T0 as above, but the radius of convergence

ˆ

of the power series possibly equals zero, we speak of a formal analytic

transformation. If in addition for some s ≥ 0 we can ¬nd constants

3.1 Formal Transformations 39

c, K > 0 so that

Tn ¤ c K n “(1 + sn), n ≥ 0, (3.2)

ˆ

we shall say that T (z) is a formal analytic transformation of Gevrey

order s. Note that being of Gevrey order s = 0 is the same as con-

ˆ

vergence of T (z) for su¬ciently large |z|.

3. Transformations of the form T (z) = diag [z p1 /q , . . . , z pν /q ], with pk ∈

Z, q ∈ N, will be called shearing transformations. If q = 1, we say

that T (z) is an unrami¬ed shearing transformation.

4. For T (z) = exp[q(z)]I, with q(z) a scalar polynomial in z, we speak

of a scalar exponential shift. Note that such a transformation is holo-

morphic at the origin, but is essentially singular at in¬nity. However,

since T (z) commutes with A(z), the transformed system has coe¬-

cient matrix A(z)’zq (z)I and hence is again meromorphic at in¬nity.

∞

5. A transformation of the form T (z) = n=’n0 Tn z ’n , with n0 ∈ Z,

will be called a meromorphic transformation if the series converges for

su¬ciently large |z|, and if in addition the determinant of T (z) is not

the zero series. Then, T ’1 (z) exists and is again of the same form. If

the radius of convergence of the series is unknown, and in particular

may be equal to zero, we speak of a formal meromorphic transforma-

ˆ

tion, and in this case write T (z) instead of T (z). If the coe¬cients

ˆ

Tn satisfy (3.2), we call T (z) a formal meromorphic transformation