0387986995Basic TheoryC

CHAPTER V

SINGULARITIES OF THE FIRST KIND

In this chapter, we consider a system of differential equations

(E)	xfy = AX, Y-),

assuming that the entries of the C"-valued function f are convergent power series in complex variables (x, y-) E Cn+I with coefficients in C, where x is a complex inde- pendent variable and y E C" is an unknown quantity. The main tool is calculation with power series in x. In §I-4, using successive approximations, we constructed

power series solutions. However, generally speaking, in order to construct a power

00

series solution y"(x) = E xmd n, this expression is inserted into the given differ-

M=1

ential equation to find relationships among the coefficients d, , and the coefficients d,,, are calculated by using these relationships. In this stage of the calculation, we do not pay any attention to the convergence of the series. This process leads us to the concept of formal power series solutions (cf. §V-1). Having found a formal power series solution, we estimate Ia',,, i to test its convergence. As the function

x-I f (x, y) is not analytic at x = 0, Theorem 1-4-1 does not apply to system (E). Furthermore, the existence of formal power series solutions of (E) is not always

guaranteed. Nevertheless, it is known that if a formal power series solution of (E) exists, then the series is always convergent. This basic result is explained in §V-2 (cf. [CL, Theorem 3.1, pp. 117-119) and [Wasl, Theorem 5.3, pp. 22-251 for the case of linear differential equations]). In §V-3, we define the S-N decomposition for a lower block-triangular matrix of infinite order. Using such a matrix, we can represent a linear differential operator

(LDO)

G[Yj = x

+ f2(x)lj,

where f2(x) is an n x n matrix whose entries are formal power series in x with coefficients in Cn. In this way, we derive the S-N decomposition of L in §V-4 and a normal form of L in §V-5 (cf. [HKS]). The S-N decomposition of L was originally defined in [GerL]. In §V-6, we calculate the normal form of a given operator C by using a method due to M. Hukuhara (cf. [Sill, §3.9, pp. 85-891). We explain the classification of singularities of homogeneous linear differential equations in §V-7. Some basic results concerning linear differential equations given in this chapter are also found in [CL, Chapter 4].

108

1. FORMAL SOLUTIONS OF AN ALGEBRAIC DE

109

V-1. Formal solutions of an algebraic differential equation

We denote by C[[x]] the set of all formal power series in x with coefficients in

w	00
C. For two formal power series f = E amxm and g =	b,,,x', we define
m=0	m=0

f = g by the condition an = b", for all m > 0. Also, the sum f + g and the

	00	00 m
product f g are defined by f +g =	(am +b,,,)xm and f g =	(F am-nbn )x"',
	m=0	m=0 n=0
	00

respectively. Furthermore, for c E C and f = >2 a,,,xm E C[[x]], we define cf

m=0

00

by cf = >camx'. With these three operations, C[[x]] is a eommutattve algebra

m=0

over C with the identity element given by E bmxm, where bo = 1 and b,,, = 0

m=0

if m > 1. (For commutative algebra, see, for example, [AM].) Also, we define the

	derivative	of f with respect to r by dx _ E(m+ 1)a,,,+,x' and the integral
				m=0
	rX	r	"0	00
			f (x)dx =	airi x"' for f = E a,,,xm E C[[xj]. Then, C[[x]]
	( f (x)dx by f
	0	o	m=1	m=0

is a commutative differential algebra over C with the identity element. There are some subalgebras of C[[x]j that are useful in applications. For example, denote by

C{x) the set of all power series in C[]x]] that have nonzero radii of convergence.

Also, denote by C[x] the set of all polynomials in x with coefficients in C . Then,

C{x} is a subalgebra of C[[x]] and C]x] is a subalgebra of C{x}. Consequently, C[x] is also a subalgebra of C[[x]].

Let F(x, yo, y, , ... , yn) be a polynomial in yo, y,..... yn with coefficients in C[[x]]. Then, a differential equation

	F	,...dxn		=0
(V.1.1)		dy	y,,d"y

is called an algebrutc differential equation, wheref y is the unknown quantity and x

is the independent variable. If F ` x, f, ... , dxn = 0 for some f in C[[x]], then such an f is called a formal solution of equation (V.1.1). In this definition, it is not necessary to assume that the coefficients of F are in C{x}.

Example V-1-1. To find a formal solution of

(V.1.2)

xLY +y-x=0,

set y = Famx'. Then, it follows from (V.1.2) that ao = 0, 2a, = 1, and

m>0

(m + 1)am = 0 ( m > 2 ). Hence, y = 2x is a formal solution of equation (V.1.2).

110	V. SINGULARITIES OF THE FIRST KIND

In general, if f E C{x} is a formal solution of (V.1.1), then the sum of f as a

convergent series is an actual solution of (V.1.1) if all coefficients of the polynomial

FareinC{x}.

Denote by x°C([x]j the set of formal series x°f (x), where f (x) E C([x]], a is a complex number, and x° = exp[a logx]. For 0 = x°f E x°C[[x]], 0 = x°g E x°Cj[x]j, and h E C(jx]], define 0 + iP = x(f + g), hO = x°hf, and

xL = ox°f + x°x . Then, x°C([xfl is a commutative differential module over

the algebra C((x]]. Similarly, let x°C{x} denote the set of convergent series x°f (x), where f (x) E C{x}. The set x°C{x} is a commutative differential module over the algebra C{x}. Furthermore, if a is a non-negative integer, then x°C([x]] C

C[[x]]. If F(x, yo,... , y,) is a formal power series in (x, yo,... , yn) and if fo(x) E xC[[x]j, ... , fn(x) E xC([xj], then F(x, fo(x),... , fn(x)) E C[[xJj. Also, if F(x, yo,. .. , yn) is a convergent power series in (x, yo,... , yn) and if fa(x) E xC{x},

... , fn(x) E xC{x}, then F(x, fo(x), ... , f,,(x)) E C{x}. Therefore, using the notation x°C[[x]]n to denote the set of all vectors with n entries in x°C[(x]], we can de-

fine a formal solution fi(x) E xC[[x]]n of system (E) by the condition x = Ax, i)

in C[[x]j . (Similarly, we define x°C{x}n.) Also, in the case of a homogeneous system of linear differential equations xfy A(x)g, a series d(x) E x°C[[xj]n is said

to be a formal solution if x = A(x)e in x°C[[x]jn. Now, let us prove the following

theorem.

Theorem V-1-2. Suppose that A(x) is an n x n matrix whose entries are formal power series in x and that A is an eigenvalue of A(O). Assume also that A + k are not eigenvalues of A(O) for all positive integers k. Then, the differential equation

(V.1.3)

x!Ly = A(x)g

has a nontrivial formal solution fi(x) = xa f (x) E x"C[[x]]n

Proof.

00

Insert g = xa E x'nam into (V.1.3). Setting A(x) _ y2 xmAm, where An E

m=0

m=0

Mn(C) and Ao = A(O), we obtain

00	Co
xa (A + m)xm= xa	LtAm_i.

Therefore, in order to construct a formal solution, the coefficients am must be determined by the equations

m-h

,1ao = Aoa"o and (A + m)am = Aodd,n + E Am-hah (m > 1 ).

h=0

1. FORMAL SOLUTIONS OF AN ALGEBRAIC DE

111

Hence, ado must be an eigenvector of Ao associated with the eigenvalue A, whereas

	m-h
ii',,, = ((A + m)1. - A0)-1	Am-hah form? 1. 0

h

For an eigenvalue Ao of A(O), let h be the maximum integer such that Ao + h is also an eigenvalue of A(O). Then, Theorem V-1-2 applies to A = A0 + h. The convergence of the formal solution ¢(x) of Theorem V-1-2 will be proved in §V-2, assuming that the entries of the matrix A(x) are in C(x) (cf. Remark V-2-9).

n	(b = x d
Let P = Eah (x)&h		/	be a differential operator with the coefficients
h=0
ah(x) in C([x]]. Assume that n > 1 and an(x) 54 0. Set
	P[x'] = x'			ab
				fm(s) xm,

m=np

where the coefficients f,,, (s) are polynomials in s and no is a non-negative integer such that 0. Then, we can prove the following theorem.

Theorem V-1-3.

(i) The degree of f,,,, in s is not greater than n.

(ii) If zeros of fno do not differ by integers, then, for each zero r of f,,, there

exists a formal series x'		(x) E xr+IC[[x]] such that P[xr(1 +O(x))] = 0.
Proof
(i) Since 6h(x'] = shx', it is evident that the degree of f,,,, in s is not greater than
n.			+o0
(ii) For a formal power series d(x) = E ckxk, we have
			k=1
		+oo
P[x'(I + b(x))] = P(x'] +		L. Ck P[x'+k J
		k=1
= x'	+oo		+oo	+oo
	(s)xm + > Ckxk (frn(s + k)xm/					}
{mnof"			k =1	m=no
= x'	fno(s)x +		(,fm(s)	+	Ckfm-k(s + k)) x"`m
		+oo	(	m-no
xa+no AK(+				k=1		J1
				+ E Ckfno+m-k(s + k)) x"' .
		M=1		k=1		J
In order that y = x''(1 +b(x)) be a formal solution of P[y] = 0, it is necessary and
sufficient that the coefficients cm and r satisfy the equations
			m
fno (r) = 0,	fno+m (r) + E Ckfno+m-k(r + k) = 0					(m > 1).

k=1

112 V. SINGULARITIES OF THE FIRST KIND

It is assumed that f,,,, (r + m) 54 0 for nonzero integer m if r is a zero of

Therefore, r and c,,, are determined by

(r) = 0 and cm = -7,(	1	m-1
		ckfn+m-k(r + k) (m > 1).
	r + m)

Convergence of the formal solution x'(1+¢(x)) of Theorem V-1-3 will be proved at the end of §V-7, assuming that the degree of in s is n and the coefficients ah(x) of the operator P belong to C{x}. The polynomial f.,, (s) is called the indicial polynomial of the operator P.

Remark V-1-4. Formal solutions of algebraic differential equation (V.1.1) as de-

fined above are not necessarily convergent, even if all coefficients of the polynomial

00

F are in C{x}. For example, y = Y (-1)m(m!)xm+i is a formal solution of the

	m=0
differential equation	2d2y
	+ y - x = 0. Also, the formal solution x' (I + fi(x)) of
	2

Theorem V- 1-3 is not necessarily convergent if the degree of f,,. (s) in s is less than

n. In order that f = >a,,,xm be convergent, it is necessary and sufficient that

m=0

la,,, I < KA' for all non-negative integers m, where K and A are non-negative num-

	00
bers. Also, it is known that some power series such as	(m!)mxm do not satisfy

'n--R

any algebraic differential equation. The following result gives a reasonable necessary condition that a power series be a formal solution of an algebraic differential equation.

Theorem V-1-5 (E. Maillet [Mail). Let F(x, yo, yi, ... , yn) be a nonzero polyno-

mial in yo, yi, ... , y,	with coefficients in C{x}, and let f = E amxm E CI[x]] be
		m=0
a formal solution of the differential equation		F(x, y, Ly, ... , dny	/	= 0. Then,
		dxn
there exist non-negative numbers K, p, and A such that
(V.1.4)	I am I < K(m!)PA'	(m > 0).

We shall return to this result later in §XIII-8.

Remark V-1-6. In various applications, including some problems in analytic number theory, sharp estimates of lower and upper bounds of coefficients am of a formal

solution f = F,00axm of an algebraic differential equations are very important.

m =9

For those results, details are found, for example, in [Mah], [Pop], [SS1], (SS2], and [SS3]. The book [GerT] contains many informations concerning upper estimates of

coefficients lam I.

2. CONVERGENCE OF FORMAL SOLUTIONS

113

V-2. Convergence of formal solutions of a system of the first kind

In this section, we prove convergence of formal solutions of a system of differential equations

(V.2.1)

Xfy = AX, Y-),

where y E C' is an unknown quantity and the entries of the C"-valued function f are convergent power series in (x, y7) with coefficients in C. A formal power series

	00
(V.2.2)	E x' ,,,, E xC([xfl"	(c,,, E C" )

rn-t

is a formal solution of system (V.2.1) if

(V.2.3)	X dx	= f(x,0)

as a formal power series.

To achieve our main goal, we need some preparations.

Observation V-2-1. In order that a formal power series (V.2.2) satisfy condition (V.2.3), it is necessary that f (O, 0) = 0. Therefore, write f in the form

f (x, y-) = o(2-) + A(x)g + F, y~'fv(x),

Iel>2

where

(1) p = (pl,... ,p") and the p, are non-negative integers,

(2) jpj = pi +

+ pn and yam' = yi'

yn^, where yi,...are the entries

of g,

(3)fo E xC{x}" and f, E C{x}",

(4)A(x) is an n x n matrix with the entries in C{x}.

Note that

fo(x) = Ax, 6) and A(x) _ Lf(x,0).

Setting

A(x) = F, x'A"s	Ao =	0,0) I
M=0		LZ

where the coefficients A", are in M"(C), write condition (V.2.3) in the form

xd'o = Ao + f (x, ¢) - A0 . dx

Then,

(V.2.4)

"t,, = A0E + rym for m = 1, 2, ... ,

114	V. SINGULARITIES OF THE FIRST KIND
where
	f (x, 0) - AoO = o(x) + IA(x) - Ao1 fi(x) +	(J(x))p
		fp(x)
	IpI>2
	00

1: xm7m

M=1

and Im E C n. Note that ry'm is determined when c1 i ... , c,,,_ 1 are determined and that the matrices mI" - A0 are invertible if positive integers m are sufficiently large. This implies that there exists a positive integers mo such that if 61, ... , C,,,a are determined, then 4. is uniquely determined for all integers m greater than mo.

Therefore, the system of a finite number of equations

(V.2.5)

mEm = A0 +

(m = 1,2,... ,mo)

decides whether a formal solution ¢(x) exists. If system (V.2.5) has a solution

{c1 i ... , c,,,a }, those mo constants vectors determine a formal solution (x) uniquely.

Observation V-2-2. Supposing that formal power series (V.2.2) is a formal solu-

	N
tion of (V.2.1), set ¢N (x) _	x'6,. Since
	m=1

Ax, (x)) - f (x, N(x)) = A(x)(d(x) - N(x)) + F, [d(x)p - N(x)D] ff(x),

Ipl>2

it follows that AX, $(x)) - f (X, dN(x)) E xN+C[Ix]]". Also,

		xd¢N(x)	= xdd(x) -	xdoN(x) E xN+1C((x((n.
		dx	dx	dx
Hence, AX, 4V W)	-	xdON(x)	E xN+1C{x}". Set
		dx
(V.2.6)		9N,0(x) =		zddN(x)
				dx

Now, by means of the transformation y" = z1+ y5N(x), change system (V.2.1) to the system

(V.2.7)	dz
	xdx = JN(x,z

on i E C", where

xdjN(x)

9N(x, l = f (x, z + dN(x)) -	dx

=9N,O(x) + f(x,Z+dN(x))f r- f(x,dN(x))

=9N,0(x) + A(x)z" + L. [(+ $N(x))" - N(x)'] f,,(x)

InI?2

2. CONVERGENCE OF FORMAL SOLUTIONS

115

As in Observation V-2-1, write gN in the form

9N(x, = 9N,O(x) + BN(x)z + L xr 9N,p(x),

Ip1?2

where

(1)9N,o E xN+1C{x}" and g""N,p E C{x}",

(2)BN(x) is an n x n matrix with the entries in C{x},

(3)the entries of the matrix BN(X) - Ao are contained in xC{x}.

Observation V-2-3. System (V.2.7) has a formal solution

			00
(V.2.8)	ON(x) = O(x) - ON(x) _		x'"cn, E xN+1C1jx11 n.
			m=N+1
The coefficients cm are determined recursively by
	me,, = Aocm + '7m	for m = N + 1, N + 2, ... ,
where
9N(x,ILN(x)) - AO1GN(x) = f(x,$(x)) - Ao1 N(x) -				dx
	= f (x, fi(x)) -		AOcN(x) - xdVx)

= 9N,O(x) + [BN(x) - Ao1 i'N(x) + E (ZGN(x))p 9N,p(x)

Ip1>2

ou

= E xm-

'1'm

m=N+1

Note that the matrices min - Ao are invertible for m = N + 1, N + 2, ... , if N is sufficiently large.

Observation V-2-4. Suppose that system (V.2.1) has an actual solution q(x)

such that the entries of i(x) are analytic at x = 0 and that ij(0) = 0. Then, the
Taylor expansion ¢(x) _	jdim (0) of q(x) at x = 0 is a formal solution of

(V.2.1). Furthermore, is convergent and E xC{x}".

Keeping these observations in mind, let us prove the following theorem.

Theorem V-2-5. Suppose that fo(x) = f (x, 0) E xN+1C{x}n and that the ma-

trices mIn - A0 (m > N + 1) are invertible, where Ao =		(0, 0). Then, system
	69
(V.2.1) has a unique formal solution
	00
(V.2.9)	t(x) = E X'n4n E x^'+1C((x}]n.

m=N+1

Furthermore, ¢ E xN+1C{x}n.

116	V. SINGULARITIES OF THE FIRST KIND

Remark V-2-6. Under the assumptions of Theorem V-2-5, system (V.2.1) pos- sibly has many formal solutions in xC[[x)J". However, Theorem V-2-5 states that there is only one formal solution in xN+1C[[xJJn

Proof of Theorem V-2-5.

We prove this theorem in six steps.

Step 1. Using the argument of Observation V-2-1, we can prove the existence and uniqueness of formal solution (V.2.9). In fact,

f (x, 4) - Ao¢ = o(x) + JA(x) - Ao}fi(x) + E (_)P_)

IpI?2

00

xmrym.

m=N+1

This implies that rym = 0 for m = 1, 2, ... , N. Hence, (m > N + 1) are uniquely determined by

mc,,, = Ao6,,, + rym for m=N+1,N+2,....

Step 2. Suppose that system (V.2.1) has an actual solution q(x) satisfying the following conditions:

(i)the entries of rl(x) are analytic at x = 0,

(ii)there exist two positive numbers K and 6 such that

jy(x)I < KIxjN+1

for

IxI < 6.

00

xm

Then, the Taylor expansion E (0) of #(x) at x = 0 is a formal solution

ml a.,.7m'1

m=N+1

of (V.2.1) (cf. Observation V-2-4). Since such a formal solution is unique, it follows that $(x) = E xm d1ij- (0). Because the Taylor expansion of if(x) at x = 0 is

m=N+1

convergent, the formal solution ¢ is convergent and E xN+1C{x}".

Step 3. Hereafter, we shall construct an actual solution f(x) of (V.2.1) that sat- isfies conditions (i) and (ii) of Step 2. To do this, first notice that there exist three positive numbers H, b, and p such that

(I)

If(x,o)I 5 HIxIN+1

for IxI < 5

and

(II)If(x,91) - f(x,y2)I 5 (IAoI + 1) 1111 - y2I

	for Ix) < b	and Iy, I	< p	(j = 1, 2).
Hence,
(III)	If(x, y-)i <_ HIxl "+1 + (IAoI + 1)Iyl for	Ixl < b	and	Iyl <_ p.

2. CONVERGENCE OF FORMAL SOLUTIONS

117

Using the transformation of Observation V-2-2, N can be made as large as we want without changing the matrix A0. Hence, assume without loss of any generality that

(V.2.10)		IAoi + 1			1
			N+1 < 2
Also, fix two positive numbers K and b so that
(V.2.11)	K > H + NAol		i	1]K	and K6N+1 < p
Step 4. Change system (V.2.1) to an integral equation
(V.2.12)				:	f n+(f ))
			J
Define successive approximations
)b(x) = 0	and	1?k+1(x) = IZ			for k=0,1,2,....
Now, we shall show that
lnk(x)I <		KIxIN+1	for		jxl <b and k=0,1,2....

Since this is true for k = 0, we show this recursively with respect to k as follows. First if this is true for k, then

I i k(x)1 < Kjxl N+l < K&N+I < p			for	jxj < b.
Hence,
I'.F(cnk (o) < {H + IIAol + 1jK}j£VN			for	6.
Therefore,
II7k+1(x)I < H + IAoI	1]K IxIN+I < K 1xIN+1			for jxj < S.
Step 5. Set	1
I1'Rk+1 -- nkll =	maxiInk+iIZI	++1k(x)I	InI < b}.

Then, since

I ik+1(x) - ik(x)I = fo

we obtain

j

IInk+1 - llkII + 171k - '1k-11I 2lI'7k - 1k-l II

N + 1

This implies that

lim ilk(x) _	#e+1(x) - #((X)
k-.+00 xN+1	xN+1
	e=0

exists uniformly for Ixl < J.