0387986995Basic TheoryC

(c) if f (x, y, µ', e) = fo (x, #,c) + A(x, µ, e)y + O(I y12), then

Let )q, A2, ... , )1n be n eigenvalues of A(oo, 0', 0) and set wf = arg A, (j _

1, 2, ... , n). Note that the w, are not unique. Set also S(pl, p2) _ {x : Pi < argx < p2}. Let a and r be two non-negative integers. Assume that the sector

S(pl, p2) is contained in the sector

32 - min{wj} + apo < (r+ 1)argx < 32 - max{wj} - apo

f o r a suitable choice of {wj : j = 1, 2, ... , n}.

Under these assumptions, we can prove the following theorem in a way similar to the proof of Theorem XIII-1-2.

¢(x, 91, e) and i (x, µ, e) such that

(i) these two solutions are bounded and holomorphic in the domain

So = {(x, 9, e) : Ix1 > N, x E S(p1, p2), 1141 < ul, 0 < IEI < e1,1 arg el < p0},

if p1 and E1 are sufficiently small,

(ii) the solution ¢(x, µ, e) admits an uniform asymptotic expansion

in So, where coefficients &(µ,e) are bounded, holomorphic, and admit uniform asymptotic expansions in powers of a as c -+ 0 in the domain {(µ, e)

Iµ1 < {31, 0 < lei < el, I argel < po},

(iii) the solution 1'(x, N, e) admits an uniform asymptotic expansion:

in So, where coefficients >Gh(x,#) are bounded, holomorphic, and admit uni- form asymptotic expansions in powers of x'1 as x - oo in the domain

{(x,µ') : Ixf > N, x E S(p1,p2), Jill < p1}.

FLrther more, there exist functions tc(x, µ, e) for t = 0,1, ... such that

(a) these functions ¢'c satisfy conditions (i) and (is) given above,

(b)

h=0

A complete proof of Theorem XIII-3-4 is found in [Si7J and (Si101.

Then, E(x) = O(x-1), F(x) = O(x'1), and Q(x) = O(x-1). Furthermore,

(XIII.4.12) becomes

Write each of three matrices E(x), F(x), and Q(x) in a block-matrix form according to that of Ao in (XIII.4.4), i.e.,

(XIII.4.15)

where EJk and QJk are nJ X nk matrices and F, are nJ x nJ matrices. Set

Substituting (XIII.4.17) into (XIrII.4.18), a system of nonlinear differential equations

is obtained. Since it is assumed that .11i ... , Al are distinct eigenvalues of Ao and that A0 is in the block-diagonal form (XIII.4.5), upon applying Theorem XIII-1-2

to (XIII.4.19) we can construct a desired holomorphic solution Qjk(x) of (XIII.4.19)

00

which admit an asymptotic expansion Qjk(x) > x "Qjk" (j, k = 1, 2,... , 3; j

"=1

k), where Qjk" are constant nj by nk matrices. Defining Fj by (XIII.4.17) and then

B(x) by (XIII.4.13), the proof of Theorem XIII-4-1 is completed. 0

Theorem XIII-4-1 concerns the behavior of solutions of system (XIII.4.1) near x = oo. Since it is useful to give a similar result concerning behavior of solutions near x = 0, we consider, hereafter in this section, a system of differential equations

where d is a positive integer and the entries of n x n matrix A(x) are holomorphic in a neighborhood of x = 0. Also, assume that A(0) is in a block-diagonal form

where a1, ... , .1t are distinct eigenvalues of A(0) with multiplicities n1, n2, ... , nt, respectively (n1 + n2 + + nt = n), and, for each j, .W is a lower-triangular and nilpotent nj x nj matrix.

Comparing the present situation with that of Theorem XIII-4-1, we notice the following two differences:

(a)singularity is at x = 0 in the present situation, while singularity is at x = 00 in Theorem XIII-4-1,

(b)The power series expansion of A(x) is convergent in the present situation, while A(x) in Theorem XIII-4-1 admits only an asymptotic expansion in a sector containing the direction arg x = 0.

We can change any singularity at x = 0 to a singularity at x = oo by changing the independent variable x by x1. Also, any direction argx = 8 can be changed

to the direction arg x = 0 by rotating the independent variable x. Furthermore, the asymptotic expansion P of Po(x) and the expansion b of Be are formal power

series satisfying the equation xd+i dP = AP - PB. This implies that two matrices

P and b are independent of P. Hence, using Corollary XIII-1-3, the following result is obtained.

Theorem XIII-4-2. Let A(x) be an n x n matrix whose entries are holomorphic in a neighborhood of x = 0. Also, let d be a positive integer. Assume that the matrix A(0) is in block-diagonal form (XIII.4.21), where al, ... , at are distinct eigenvalues of A(0) with multiplicities n1, n2, ... , nt, respectively (n1 + n2 +, + nt = n), and for each j, JVj is a lower-triangular and nilpotent n, x nj matrix. Fix a real number

0 so that (a, - Xk)e-utO V EP for j 96 k. Then, there exists two positive numbers be and ee and an n x n matrix Pe(x) such that

(a) the entries of Po(x) are holomorphic and admit asymptotic expansions in pow-

00

(b) if >2 xmP,,, is the asymptotic expansion of Pe(x), then this expansion is

m=0

independent of 8 and Po = I,,,

(c) the transformation PB(x)u" changes system (XIIL4.20) to a system

For each,, Bje is an n, xn2 matrix which admits also an asymptotic expansion in x as x--i0 in So.

The main claims of this theorem are

424 XIII. SINGULARITIES OF THE SECOND KIND

(i) the asymptotic expansion of Pe(x) is independent of 0,

Proof of Theorem XIII-4-2 is left to the reader as an exercises.

Remark XIII-4-3. Using Theorem XIII-3-4, we can generalize Theorem XIII-

4-1 to the system e°dx = xr'A(x, µ, e)y", where r and o are non-negative integers,

y" E C^, and A(x, µ', e) is an n x n matrix with the entries that are bounded and holomorphic with respect to the variable (x, µ, e) E C x C'° x C in a domain

Do = {(x, µ, e) : 1x1 > N, Iµ1 < µo, 0 < 1e1 < eo, 0 < 1 arg e1 < po}. Also, assume

00

that A admits a uniform asymptotic expansion A(x, µ, e) - > ehAh(x, µ) in Do

h=0

as a -+ 0, where coefficients Ah (X, µ) are bounded and holomorphic in the domain

{(x,µ) : 1x1 > N, 1µ1 < µo}. A complete proof of this result is found in [Si7] and

[Hs2].

XIII-5. Cyclic vectors (A lemma of P. Deligne)

In the study of singularities, a single n-th-order differential equation is, in many cases, easier to treat than a system of differential equations. In this section, we explain equivalence between a system of linear differential equations and a single n-th-order linear differential equation.

Let us denote by 1C the field of fractions of the ring C[(x]] of formal power series in x, i.e.,

Also, denote by V the set of all row vectors (cl(x), c2(x), ... , c,,(x)), where the entries are in the field K. The set V is an n-dimensional vector space over the

field 1C.

Define a linear differential operator C : V - V by G[vl = by + 7l(x) (v' E V ), where b = x d and S2(x) is an n x n matrix whose entries are in the field 1C. We

first prove the following lemma.

Lemma XIII-5-1 (P. Deligne [Del]). There exists an element iio E V such that

{v"o, Gv"o, G2v"o, ... , G"-lvo} is a basis for V as a vector space over IC.

Proof.

For each nonzero element v of V, denote by µ(v'' the largest integer t such that

{ii, Cii, C2v, ... ,.CeV) is linearly independent over K. In two steps, we shall derive

a contradiction from the assumption that max{µ(v') : v' E V} < n - 1.

Step 1. First, we introduce a criterion for linear dependence of a set of elements of V. Consider a set {i11, ... vm } C V, where m is a positive integer not greater than n = dim, V. Let 6j = (c31,c12,... ,Cin) (j = 1,2,... ,m). Set .7 = {(jl,... ,jm)

1 < it < j2 < . < jm < n}, and introduce a linear order .7 -' { 1, 2,... , (M-)) in the set J. Let us now define a map

V ={(v1,V2,...,Vm): 61 EV (j=l,...,m)} -+ (.) m Id.)

(VI, ... , Vm) --+ V1 A v2 A ... A vm, where

1,AV2A ... A vm

(4) a sety{ {6j, 62,... , Vm } E V is linearly dependent if and only if v'1 AV2 A A 17m =

6 in K('^).

Step 2. Fix an element Vo of V such that µ(6o) = max{p(V) : v" E V} < n-1. Since p(VO) < n-1, another element w of V can be chosen so that {VO, Cv"o, CZi3o, ... , CnOv', w"} is linearly independent, where no = max{µ(v") : v E V}. Set v' = vo+Ax"'O E V, where A E C and m is an integer. Then, Cwt = Cw"o + C'(Ax-ti) = Crvo +

Axm(C + m)tw. Since {V, Cv", ... , Cn0+1V} is linearly dependent, it follows that A CnOv' A C"O+I V = 6 for all A E C and all integers in. Note that

A C"OiYA C"°+16 is a polynomial in A. Since this polynomial is iden- tically zero, each coefficients must be zero. For example, the constant term of this polynomial is i6o A CVO A ... A 00+I V"o. This is zero since {iio, Ciio..... G"0+Iuo} is linearly dependent. Compute the coefficient of the linear term in A of the polynomial. Then,

no

+ 1:60A...ACi-IVoA (C+m)lw A CJ+'A A...ACnb+Ii = 0

J=1

identically for all integers m. The left-hand side of this identity is a polynomial in in of degree no + I. Hence, each coefficient of this polynomial must be zero. In particular, computing the coefficient of mn0+1, we obtain vOAGvOA . A GnbtloAw' =

0. This is a contradiction, since {vo, Gvo, ... , L"Ovo, w} is linearly independent.

This completes the proof of Lemma XIII-5-1.

Definition XIII-5-2. An element vo E V is called a cyclic vector of G if {6o, Lu0,

L2v'o, ... , Ln-1%) is a basis for V as a vector space over X.

Observation XI II-5-3. Let uo be a cyclic vector of Land let P(x) be the n x n ma-

=and, hence, setting A(x) = L[P(x)]P(x)-1 = bP(x)P(x)-l +

P(x)f2(x)P(x)-1, we obtain

with the entries a, E C. Thus, we proved the following theorem, which is the main result of this section.

Theorem XIII-5-4. The system of differential equations

if y is changed by u = P(x)y". System (XIll.5.2) is equivalent to the n-th-order differential equation

1=o