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Proof: First, assume that (1.6) has a singularity of the п¬Ѓrst kind at the
origin. Then the equivalent system constructed in Exercise 1 has a singu-
larity of the п¬Ѓrst 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 Вµ , with Вµ в€€ 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. Redeп¬Ѓning Вµ mod 1, we therefore may assume
в€ћ
|z| < ПЃ,
yn z n+Вµ ,
y(z) =
n=0

with y0 = 0. As for systems, we call such a solution y(z) a Floquet solution
of (1.6). Making ПЃ smaller if necessary, we may also assume that y(z) does
not vanish on R(0, ПЃ). We now complete the proof by induction with respect
to ОЅ: For ОЅ = 1, we obtain from (1.6) that z a1 (z) = zy (z)/y(z), and on the
right one can cancel z Вµ . Thus z a1 (z) is the quotient of two holomorphic
functions, with a nonzero denominator, hence holomorphic at the origin.
So for ОЅ = 1 we completed the proof. For larger ОЅ, we substitute y = y(z)Лњ 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) в‰Ў в€’1):
ОЅв€’1 ОЅ
k y (kв€’j) (z)
(ОЅ) (j)
y
Лњ = y
Лњ aОЅв€’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 ОЅ в€’1. Observe that the factor z Вµ can
Лњ
(kв€’j)
be canceled from y (z)/y(z), which then becomes single-valued at the
origin, with a pole of order j в€’k. So the coeп¬ѓcients of this new equation are
holomorphic in R(0, ПЃ). By assumption, no solution y (hence: no y , or its
Лњ
derivative) can grow faster than powers of z, so by induction hypothesis,
the new ODE has a singularity of the п¬Ѓrst kind at the origin. Thus we
conclude that the coeп¬ѓcient of y (j) can at most have a pole of order ОЅ в€’ j
Лњ
at the origin. This information can be used to conclude by induction that
each aОЅв€’k (z) can at most have a pole of order ОЅ в€’ k. 2
According to Exercise 3, one can always recursively compute the coeп¬ѓ-
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 ОЅ в€’ 1. This, however, is not a very eп¬ѓcient
procedure.
The Floquet exponents Вµ 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
Вµ
sk (z) [log z]k ,
y(z) = z
k=0

where Вµ 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 п¬Ѓnitely many power series coeп¬ѓcients of these functions goes
under the name FrobeniusвЂ™ method. In the exercises below we have described
an essential part of this method; for the cases not included there, see .
In any case, for ОЅ в‰Ґ 3, one will only in exceptional cases succeed in п¬Ѓnding
all power series coeп¬ѓcients of the functions sk (z) in closed form.

Exercises: For the following exercises, let a ОЅth-order linear ODE (1.6)
in G = R(0, ПЃ), ПЃ > 0, be given.

1. Deп¬Ѓning x = [y, zy , . . . , z ОЅв€’1 y (ОЅв€’1) ]T , and
пЈ® пЈ№
0 1 0 ... 0 0
пЈЇ0 пЈє
1 1 ... 0 0
пЈЇ пЈє
пЈЇ0 пЈє
0 2 ... 0 0
пЈЇ пЈє
A(z) = пЈЇ . пЈє,
. . . .
..
пЈЇ. пЈє
. . . .
.
. . . . .
пЈЇ пЈє
пЈ°0 пЈ»
... ОЅ в€’ 2
0 0 1
bОЅ (z) bОЅв€’1 (z) bОЅв€’2 (z) . . . b2 (z) b1 (z) + ОЅ в€’ 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 п¬Ѓrst kind at the origin, i.e.,

b(k) z n , |z| < ПЃ, 1 в‰¤ k в‰¤ ОЅ.
bk (z) = z k ak (z) = n
nв‰Ґ0

Let y(z) = n yn z n+Вµ , y0 = 0, Вµ в€€ C , be a solution of (1.6). Show
that then the Floquet exponent Вµ satisп¬Ѓes the indicial equation
ОЅ
(k)
[Вµ]ОЅ = b0 [Вµ]ОЅв€’k ,
k=1

with [Вµ]0 = 1, [Вµ]k = Вµ(Вµ в€’ 1) В· . . . В· (Вµ в€’ k + 1), for k в€€ N.
3. For bk (z) and y(z) as above, п¬Ѓnd the equations that the coeп¬ѓcients
yn have to satisfy so that y(z) is a solution of (1.6). Show that for
at least one solution Вµ of the indicial equation one can recursively
compute the coeп¬ѓcients 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)
ОЅ
(a) With p0 (w) = [w]ОЅ в€’ k=1 b0 [w]k , show that the inhomoge-
neous equation
ОЅ
ОЅ (ОЅ)
z ОЅв€’k bk (z) y (ОЅв€’k) = p0 (w) z w
в€’
zy
k=1

в€ћ
has a unique solution y(z; w) = 0 yn (w) z n+w , with y0 (w) в‰Ў 1
and coeп¬ѓcients yn (w), n в‰Ґ 1, which are rational functions of w.
Find the possible poles of pn (w) in terms of the roots of p0 (w).
(b) Let Вµ be a root of p0 (w), but so that p0 (Вµ + j) = 0 for j в€€ N.
Conclude that then y(z; Вµ) is a Floquet solution of (1.6). Show
that for at least one root of p0 (w) this assumption holds.
(c) Let Вµ, as above, be a root of multiplicity at least k в‰Ґ 2. Show
that then (1.6) has a solution of the form
в€ћ
kв€’1
kв€’1 (j)
|z| < ПЃ.
[log z]kв€’1в€’j yn (Вµ) z n+Вµ ,
j n=0
j=0

Show that all solutions so obtained are linearly independent.
5. For the hypergeometric, resp. BesselвЂ™s, equation, п¬Ѓnd 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 п¬Ѓrst kind, we may without loss of generality assume that z0 is any
preassigned point in C в€Є {в€ћ}. Both for historical reasons as well as nota-
tional convenience it is customary to choose z0 = в€ћ here. Hence we shall
be dealing with systems of the form
в€ћ
An z в€’n , |z| > ПЃ,
r
z x = A(z) x, A(z) = z (3.1)
n=0

with PoincarВґ rank r в‰Ґ 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 PoincarВґ 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 inп¬Ѓnity. Expanding
в€ћ
S(z) = в€’в€ћ Sn z в€’n , inserting into (3.1), and comparing coeп¬ѓcients gives
в€ћ
Snв€’r (M в€’ (n в€’ r)I) = n в€€ Z.
Am Snв€’m ,
m=0

This is a homogeneous system of inп¬Ѓnitely many equations in inп¬Ѓnitely
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 п¬Ѓrst
kind we have seen that S(z) cannot have an essential singular point at
inп¬Ѓnity; hence Sn = 0 for small n в€€ 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 )в€ћ for which S(z) converges and
в€’в€ћ
det S(z) = 0 for suп¬ѓciently large |z|, to п¬Ѓnd M and (Sn ) is much more
diп¬ѓcult. There is a theory of inп¬Ѓnite determinants (see von Koch )
that might be applied here. We shall not do this, however, because even
if we had computed M and the coeп¬ѓcients Sn , we would not obtain any
detailed information on the behavior of solutions for z в†’ в€ћ. 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 inп¬Ѓnity 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 п¬Ѓrst 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 , Kimura , Malgrange [182, 183], Hukuhara [131, 132],
Bertrand , and Varadarajan [274, 275]. For presentations using more
algebraic tools, see Gerard and Levelt , Levelt , or Babbitt and

3.1 Formal Transformations
Since we are now working in a neighborhood of inп¬Ѓnity, 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)Лњ employed in this chapter:
x

в€ћ
1. If T (z) = 0 Tn z в€’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)Лњ, an analytic transformation (at the point
x
inп¬Ѓnity). This coincides with the deп¬Ѓnition in Section 2.4, up to the
change of variable z в†’ 1/z. If only п¬Ѓnitely many coeп¬ѓcients Tn are
diп¬Ђerent from zero, we say that the transformation terminates.
в€ћ
2. If T (z) = 0 Tn z в€’n , with T0 as above, but the radius of convergence
Л†
of the power series possibly equals zero, we speak of a formal analytic
transformation. If in addition for some s в‰Ґ 0 we can п¬Ѓnd constants
3.1 Formal Transformations 39

c, K > 0 so that

Tn в‰¤ c K n О“(1 + sn), n в‰Ґ 0, (3.2)
Л†
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-
Л†
vergence of T (z) for suп¬ѓciently large |z|.
3. Transformations of the form T (z) = diag [z p1 /q , . . . , z pОЅ /q ], with pk в€€
Z, q в€€ N, will be called shearing transformations. If q = 1, we say
that T (z) is an unramiп¬Ѓed 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 inп¬Ѓnity. However,
since T (z) commutes with A(z), the transformed system has coeп¬ѓ-
cient matrix A(z)в€’zq (z)I and hence is again meromorphic at inп¬Ѓnity.
в€ћ
5. A transformation of the form T (z) = n=в€’n0 Tn z в€’n , with n0 в€€ Z,
will be called a meromorphic transformation if the series converges for
suп¬ѓciently large |z|, and if in addition the determinant of T (z) is not
the zero series. Then, T в€’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-
Л†
tion, and in this case write T (z) instead of T (z). If the coeп¬ѓcients
Л†
Tn satisfy (3.2), we call T (z) a formal meromorphic transformation
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