0387986995Basic TheoryC

M2 (x) = 1 o + xvl + x2v2

and

Thus, we arrived at the following conclusion.

Furthermore, the transformation y" = P(x)Po iv changes (V.5.20) to

In the case when a=1, wefixm=2. Then

Q2(x) = Po + xP1 + T2 p2.

+oo

Conclusion V-5-11. There exists a unique 2 x 2 matrix P(x) = >xtPt such

e=o

that

(1)the matrices Pt for P = 0, 1, 2 are given by (V.5.23),

(2)the power series P(x) convet es for every x,

(3)the transformation y = P(x)u changes the system

For a further discussion, see IHKS[. A computer might help the reader to calculate

S2i N2, and P2 in the cases when a = 0 and a = 1. Such a calculation is not difficult in these cases since eigenvalues of A2 are found easily (cf. §IV-1).

V-6. Calculation of the normal form of a differential operator

In this section, we present another proof of Theorem V-5-1. The main idea is to construct a power series P(x) as a formal solution of the system

dP(x) = P(x)(Ao + vo(x)) - Q(x)P(x)

(cf. Remark V-5-2).

Another proof of Theorem V-5-1.

To simplify the presentation, we assume that So = A0 = diag[,ulI,, µ2I2, ... ,

Aklk[, where pi, p2, ... , µk are distinct eigenvalues of no with multiplicities m1, respectively, and the matrix I, is m, x m j identity matrix. Since AOA(o = NoAo,

the matrix No must have the form No = diag(N01,No2, ... ,Nok[, Where Nor is an

00

mi x m1 nilpotent matrix. Let us determine two matrices P(x) = I,a + E xmP,,,

M=1

[=1

Therefore, it suffices to solve the equation

(V.6.1) mX + (A0 +N4)X - X(Ao+No) - Y = H,

where X and Y are n x n unknown matrices, whereas the matrix H is given. If we write X, Y, and H in the block-form

where XXh, YYh, and H,h are m, x mh matrices, equation (V.6.1) becomes

(m + P) - µh)X)h + NO,Xjh - X,hNOh - Yjh = Hjh,

where j, h = 1,... , k. We can determine XJh and Y,h by setting Y,;h = 0 if m + u, - Ah 4 0, and X,,h = 0 if m + p, - µh = 0. More precisely speaking, if m + p - Ph 96 0, we determine X3h uniquely by solving

(m + p, - µh)Xlh + NojXjh - X3hNoh = H)h.

If M + Aj - Ah = 0, we set Y,h = H,h. In this way, we can determine P(z) and

B(x). In particular, go + B(x) has the form x ^°I'xA0, where r is a constant

n x n nilpotent matrix. Furthermore, the operator x d +A0 and the multiplication

ds

operator by No + B(x) commute. D

The idea of this proof is due to M. Hukuhara (cf. [Si17, §3.9, pp. 85-891).

Remark V-6-1. In the case of a second-order linear homogeneous differential equation at a regular singular point x = a, there exists a solution of the form

+00

01(z) = (x - a)aE c, (x - a)n, where the coefficients c,, are constants, co 0 0,

n=O

and the power series is convergent. If there is no other linearly independent solution of this form, a second solution can be constructed by using the idea explained in Remark IV-7-3. This second solution contains a logarithmic term. Similarly, a third-order linear homogeneous differential equation has a solution of the form

+00

01(x) _ (x - a)° c, (x - a)" at a regular singular point x = a. Using this solu-

n=0

tion, the given equation can be reduced to a second-order equation. In particular, if there exists another solution ¢(x) of this kind such that ¢1 and ¢2 are linearly independent, then the idea given in Remark IV-7-4 can be used to find a fundamental set of solutions.

In general, a fundamental matrix solution of system (V.5.14) can be constructed if the transformation y" = P(x)xLV of Theorem V-5-4 is found. In fact, if the

matrix solution of (V.5.14). The matrix P(x) can be calculated by using the method of Hukuhara, which was explained earlier.

V-7. Classification of singularities of homogeneous linear systems

In this chapter, so far we have studied a system

where the entries of the n x n matrix A(r) are convergent power series in x with complex coefficients. In this case, the singularity at x = 0 is said to be of the first kind. If a system has the form

where k is a positive integer and the entries of the n x n matrix A(x) are convergent power series in x with complex coefficients, then the singularity at x = 0 is said to be of the second kind

In §V-5, we proved the following theorem (cf. Theorem V-5-4).

Theorem V-7-1. For system (V.7.1), there exist a constant n x n matrix A0 and an n x n invertible matrix P(x) such that

(i)the entries of P(x) and P(x)-1 are analytic and single-valued in a domain

0 < jxi < r and have, at worst, a pole at x = 0, where r is a positive number,

(ii) the transformation

Theorem V-7-1 can be generalized to system (V.7.2) as follows.

7. CLASSIFICATION OF SINGULARITIES OF LINEAR SYSTEMS 133

Theorem V-7-2. For system (V.7.2), there exist a constant n x n matrix A0 and an n x n invertible matrix P(x) such that

(i)the entries of P(x) and P(x)-1 are analytic and single-valued in a domain

V = {x : 0 < lxi < r}, where r is a positive number,

(ii)the transformation

changes (V.7.2) to (V.7.4).

Proof.

Let 4i(x) be a fundamental matrix of (V.7.2) in D. Since A(x) is analytic and single-valued in D, fi(x) _ I (xe2"t) is also a fundamental matrix of (V.7.2). Therefore, there exists an invertible constant matrix C such that 46(x) = 4i(x)C (cf. (1) of Remark IV-2-7). Choose a constant matrix Ao so that C. = exp[2rriAo] (cf. Ex- ample IV-3-6) and let P(x) = 4(x)exp[-(logx)Ao]. Then, P(x) and P(x)-1 are analytic and single-valued in D. Furthermore,

dP(x) = dam) exp(_(logx)Ao] - P(x)(x-'Ao)

= x-(k+1)A(x)P(x) - P(x)(x-'Ao).

This completes the proof of the theorem. 0

An important difference between Theorems V-7-1 and V-7-2 is the fact that the matrix P(x) in Theorem V-7-2 possibly has an essential singularity at x = 0.

The proof of Theorem V-7-2 immediately suggests that Theorem V-7-2 can be extended to a system

dx

where every entry of the n x n matrix F(x) is analytic and single-valued on the domain V even if such an entry of F(x) possibly has an essential singularity at x = 0. More precisely speaking, for system (V.7.6), there exist a constant n x n matrix Ao and an n x n invertible matrix P(x) satisfying conditions (i) and (ii) of

Theorem V-7-2 such that transformation (V.7.5) changes (V.7.6) to (V.7.4). Now, we state a definition of regular singularity of (V.7.6) at x = 0 as follows.

Definition V-7-3. Let P(x) be a matrix satisfying conditions (i) and (ii) such that transformation (V.7.5) changes (V.7.6) to (V.7.4). Then, the singularity of

(V.7.6) at x = 0 is said to be regular if every entry of P(x) has, at worst, a pole

atx=0.

Remark V-7-4. Theorem V-7-1 implies that a singularity of the first kind is a regular singularity. The converse is not true. However, it can be proved easily that a regular singularity is, at worst, a singularity of the second kind. Furthermore, if (V.7.2) has a regular singularity at x = 0, then the matrix A(0) is nilpotent. This

is a consequence of the following theorem.

Theorem V-7-5. Let A(x) and B(x) be two n x n matrices whose entries are formal power series in x with constant coefficients. Also, let r and s be two positive integers. Suppose that there exists an n x n matrix P(x) such that

(a)the entries of P(x) are formal power series in x with constant coefficients,

(b)det(P(x)] ,E 0 as a formal power series in x,

B(x)il.

Suppose also that s > r. Then, the matrix B(O) must be nilpotent.

Proof

Step 1. Applying to the matrix P(x) suitable elementary row and column opera- tions successively, we can prove the following lemma.

Lemma V-7-6. There exist two n x n matrices

and n integers 1\1,,\2, ... , an such that

(i)the entries of n x n matrices T,,, and Sm are constants.

(ii)det To 0 0 and det So 0 0,

(iii)

T(x)P(x)S(x) = A(z) = diag(x''',zA2,...

(iv)A1<A2<...<An .

The proof of this lemma is left to the reader as an exercise.

Step 2. Change the two systems x'd = A(x)y and x'd6 = B(x)u, respectively,

to xr d = C(x)a and x' d = D(x)ii by the transformations z' = T(x)y and

ii = S(x)v. Then, F = A(x)v. Furthermore, if the matrix D(0) is nilpotent, the matrix B(0) is also nilpotent.

Step 3. Look at D(x) = x'-'A(x)-1C(x)A(x) - x'A(x)-ld ). This shows clearly that if s > r, the matrix D(0) is nilpotent.

The following corollary of Theorem V-7-5 is important.

Corollary V-7-7. Assume that conditions (a), (b), and (c) of Theorem V-7-5 are satisfied. Also, assume that A(O) and B(O) are not nilpotent. Then, r = s.

Definition V-7-8. If a singularity of the second kind is not a regular singularity, this singularity is said to be irregular. In particular, if the matrix A(O) of (V.7.2) is not nilpotent, the singularity of (V.7.2) at x = 0 is said to be irregular of order k. Also, a regular singularity is said to be of order zero.

In order to define the order of singularity at r = 0 for all systems (V.7.2), the

independent variable x must be replaced by x'1P with a suitable positive integer p.

assume that the coefficients ah are convergent power series in x and that an (x) # 0. Set y; = 6'-1y (j = 0,... , n - 1). Then, (V.7.7) becomes the system

Set bh(x) = ah(x)x(n-h)O . Choose a non-negative rational number a so that bh(X) an(x)

(h = 0,... , n - 1) are bounded in a neighborhood of x = 0. Also, if a must be positive, choose a so that bh(0) 0 0 for some h. Then, the matrix on the right-hand side of (V.7.9) is not nilpotent at x = 0 if or > 0. Hence, we define a as the order of the singularity of (V.7.7) at x = 0. System (V.7.2) can be reduced to an equation

(V.7.7) (cf. §XIII-5).

The following theorem due to J. Moser [Mo) concerns the order of a given sin- gularity-

Theorem V-7-9. Let A(x) = x-a E x'A be an n x n matriz, where p is an

=o

integer, A are n x n constant matrices, A0 y6 0, and the power series is convergent.

Set

Im(A) = max (P_ 1 + n,0) ,

µ(A) = minlm(T-1AT-T-1dx)J

(ii)In the case when the degree of fn ins is less than n, if we change the equation

P[y] = 0 by setting y1 = y and y, = b'-1 [y] (j = 2,... , n), the system has an irregular singularity at x = 0.

Proof

ah(x) = xnobh(x). Then, the functions bh(x) are analytic at x = 0. Furthermore,

write the equation P(y) = 0 in the form bo[y] = -bn(Ebh(x)bh(yJ. Claim (i)

h=0

follows immediately from this form of the equation.

If the degree of fn0 is less than n, we must have bn(0) = 0 and b,(0) - 0 for

with a suitable positive rational number a so that the system for (z1, ... , zn) has a singularity of the second kind of a positive order (cf. the arguments given right after Definition V-7-8, and also §XIII-7).

From Theorem V-7-10, we conclude that the formal solution x''(1 + O(x)) of

Theorem V-1-3 is convergent if the degree of f,, (s) in s is n. The following corollary of Theorem V-7-10 is a basic result due to L. Fuchs [Fu].

Corollary V-7-11. The differential equation Ply] = 0 has, at worst, a regular

singularity at x = 0 if and only if the functions ah(x) (h = 0, ... , n - 1) are x

analytic at x = 0.

Some of the results of this section are also found in [CL, Chapter 4, pp. 108-137].

EXERCISES V

V-1. Show that if A is a nonzero constant, H is a constant n x m matrix, and N1 and N2 are n x n and m x in nilpotent matrices, respectively, then there exists one and only one n x m matrix X satisfying the equation AX + NIX - XN2 = H.

V-2. Show that the convergent power series

where a, /3, and -y are complex constants.