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JOURNAL OF MATHBMATICAL ANALYSIS AND APPLICATIONS 16, 84-103 (1966)
On the Asymptotic Integration of Second Order Linear
Ordinary Differential Equations with Polynomial Coefficients
PO-FANG HSIEH AND YASUTAKA SIBUYA*
Department of Mathematics, Western Michigan University, Kalamazoo, Michigan; and School of Mathematics, University of Minnesota, Minneapolis, Minnesota Submitted by Gian Carlo Rota
1. INTRODUCTION
Let a differential equation
y”-P(x)y=O (^) (#=y&) (1.1)
be given, where P(X) is a polynomial of X:
P(x) = xm + alxm-1 + **a + a,-,~ + a,,
and the coefficients aj are complex parameters.
The only singular point of equation (I. 1) is x = co. Therefore a solution of (1.1) is an entire function of (x, a, , ..a, a,) if its initial values are entire functions of (a, , ***, a,).^ On^ the other^ hand, since x =^ COis an irregular singular point, we can determine a solution of equation (1.1) by prescribing asymptotic conditions as x tends to infinity in a sector 9, if 9’ and the asymptotic conditions at x = co are suitably given. Analytic properties of such a solution with respect to parameters are important in the study of boundary value problems for Eq. (1.1). In this paper, we shall prove the following:
THEOREM 1. The dtflerential equation y” - P(x) y = 0, where
P(x) = xm + aIxm--l + -0. + a,-9 + a,
and the coe#cients aj are complex parameters, has a solution
Y = gdx, a1 y a2, --) a,) (1.3)
- This work was partially supported by National Science Foundation (GP 3904) and was partially based on Mathematics Research Center, United States Army, University of Wisconsin, Technical Summary Report, No. 505, September, 1964. 84
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS
such that
(i) gm is an entire function of (x, a, , **-, a,); (ii) ?V,,,and g,,,’ admit respectively the asymptotic representations
gm E xrm 1 + f B&&j I (^) N=l
X exp - 2(m + 2)-l x*(~+~) + I
m+ & &,,~~‘“+~-~‘1 ,
Ym’ g - X++‘m 1 + 2 C&N) I (^) N- m+ x exp I
- 2(m + 2)-l .&(m+2) + c &p+(m+2--N) N=l I
uniformly on each compact set in the (a, , a.., u&space as x tends to infinity in any closedsector which is contained in the sector
1arg x 1 < 3(m + 2)-l T, (^) U-6) where Y, , Am,N , Bm,N and C,,N are polynomials of a, , ***, a,,, , and x’=exp(r(logIxI +iargx)} (^) (1.7)
for any constant r.
The proof of this theorem will be given in Sections 6-13. In Sections 2- we shall explain some properties of the solution g,,,.
REMARK 1. Let us put
P(x)=(x-Al)(x-A2)~-(x-Ah,), (^) ( where A, , .a., h, are the roots of the polynomial P(X). Since a,, e-e,unz are symmetric polynomials of A, , me.,A, , the solution^ g,,, is an entire^ function of (X,hr, ., A,) which is symmetric with respect to (A1, a,A,), while Y, , A m.N > %.N and crnN are symmetric^ polynomials^ of (A, , q-v,A,).
- COMPUTATION OFI,, Am-N, Bm,N AND Cm.,
Let Y&x, a,, se*,a,) denote the right-hand member of (1.4), and put
I
m+ B,(x, a, , e, a,) = x-‘m exp 2(m + 2)-l x*(~+~) - c Am,N~+(m+2-N) Y,. N-1 I (2.1)
where
By putting
we get
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS
I
for odd m,
s, =
--:Ltmtl for even^ m.
2 = Ym{l + 0(x-f)},
Y
P-8)
Now let us put 2 = X’“W. (2.10) Then equation (2.6) becomes
wn + 2(Y,x-1 + )h) w’
- {(rmx-1)2 - 2Y,x-2 + 2;f;rmx-’ +/z’ + j” - P(x)} w = 0. (2.11)
The coefficient of w’ is x*~(-- 2 + 0(x-l)}, while the coefficient of w is of O(X~+~). Hence by putting
w = 1 + f B,,,,N~-+N, (2.12) N-l
we can determine Bm,N as polynomials of a, , me*,a,,,.
&MARK 2. In the case when m is even, the asymptotic representation of g,,, does not contain any fractional powers of x-l.
- THE SOLUTIONS 9Ym,k
Let us change the independent variable x by
ff = eiex.
Then equation (1.1) becomes
--e^ d”Y
di?
--t(tn+2)8 Q(4 y = 0,
where
88 HSIEH AND SIBUYA
Therefore, if we choose 6 so that ei(m+2)e= 1, the function
Ym(i, eieal , ei20a3 , **a, eimea,)
is a solution of Eq. (3.2). Hence if we put
l& = 2&n + 2)-i 7r (h = 0, 1, -*a, m + 1) and gm.dx, a1 , --, a,)^ =^ Ym(eisk x, eiek a,^ , eiwk a2 , -a-, eimeka^ WZ,)
then Ym,k are solutions of Eq. (1.1). Thus we obtained the
THEOREM 2. The solution Ym,k satisfies^ the following^ conditions: (9 Ym.k is^ an^ entire function^ of (x, al^ , e-e,^ a,); (ii) Ym,k and YA,k admit respectively the asymptotic representations
Ym,k g Ym(eiek x, eiek a, , eizek a2 , a, eimeka,). (^) (3.6)
Y& g eiek Y,,,‘(eie, x, eiek a, , a**, ei”“I a,) (^) (3.7)
uniformly on each compact set in the (a, **a, a,)-space as x tends to infinity in any closed sector which is contained in the sector
I argx + 0, I < 3(m + 2)-ln (3.8)
REMARK 3. It is evident that Y,,, = Ym,s.
REMARK 4. Ym,* are also symmetric functions of (A,, a,L). Since Q(i) = (2 - eieAl) (2 - eieA2)* (2 - eieAm), (^) (3.9)
we get YW,k from Ym by replacing (x, A, , s-e,&J by
(eiekx, eiekAl, *a- , eiekAJ.
4. SUBDOMINANT SOLUTIONS AND UNIQUENESS OF Y,,,
Let Sp, denote the sector in the x-plane which is defined by
1arg x + B,I < (m + 2)-l r,
and let & denote the closed sector defined by
j arg x + 8, 1 < (m + 2)-l W.
90 HSIEH^ AND SIBUYA
From the fact that
I
1 + 5 ahx-h
h=l 1
1 tq
I
1-f h=l
1+f eihebh(eisx)-~, h=l
1 + -f b,x-” h=l
it follows that
and
btm+l(etea, , efzea, , e-e, eimsa,) = ei’+m+l)e btm+l(a, , a-*, a,) (^) (4.7)
b(eiex, eieal , eizea2 , e-0,eimeam) = eii(m+2)e b(x, a, , *-a, a,).
Since ei*tm+s)sr = - 1, we obtain
(4.8)
where
W,,,(a, , --a, a,) = 2eie1Pm, (^) (4.9)
I
-*m for odd m, Pm = -im+bti+l for even^ m.^
(4.10)
- COMPUTATIONOF?VmASA POWERSERIESOF(al , a.., a,)
The solution ?Y,,,is an entire function of (x, a, , a, a,). Therefore this function can be represented as a power series of (al , a-3,a,) with coefficients that are entire functions of x. This power series is uniformly convergent on each compact set in the (x, a, , ..., a,)-space. Put
g&, al, --9 am) = 7,&) + C’ a? a*a~~7m.v, .. v,,(x) , (5.1)
where p, , s-s,p, are^ non-negative^ integers,^ the^ quantities^ ~,&x)^ and
%n.v,-vm(x) are entire^ functions^ of x and Z’^ =^ ZD,+...+vm,,>l. Let^ us put
Y&G al , -*, a,) = L.,(x) + C’ a? *** &%m.v, *** ,(4. (5.2)
Since r,,, , Am,N and B,,,N are polynomials of (a, , e., a,), the right-hand member of (5.2) is well defined and the quantities [,,,, and I,,,,..., are formal series consisting of exponentials and powers of log x, zcand X-*.
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS 91
It is easy to prove that
and
as x tends to infinity in any closed sector which is contained in the sector (1.6). Let us insert the power series (5.1) into Eq. (1.1). Then we obtain the following system of equations:
and
where Km,,l.a.,)m(x) is a function of x and r]“,,l...,m(x) such that
kilqk = ,j$ - I*
The quantities v,,&x) and ~~,,&x) are determined uniquely by Eqs. (5.5) and (5.6) and asymptotic conditions (5.3) and (5.4). In particular, T,,,(X) is a constant multiple of
E”H,‘1’(0 (5.7)
where H;‘(t) is the Hankel function of the first kind of the order v and
[ = 2i(m + 2)-l x+(*+2), v = (m + 2)-l.
The constant multiplier can be found by comparing the quantity [,,Jx) with the asymptotic representation of (5.7). In order to determine ~,,,~,...,~(x) we may use the method of the variation of parameters.
6. PROOF OF THEOREM 1: PART I
Now we shall prove Theorem 1 in several steps. As the first step, we shall reduce Eq. (1.1) to a system of equations. Put iY.= Y [IY’.
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS 93
7. PROOF OF THEOREM 1: PART II
We shall now derive a nonlinear differential equation associated with system (6.3), and state a fundamental lemma concerning this associated nonlinear equation. Let us put fG> 81(4> B(h) = b2cn a2(5)I.
The quantities CX~and & are linear in a, , a*-, a, and polynomials in E-1, and we have % = - 2 + O(P), Bl = O(t a2 = 2 + O(P), B2 =^ O(P).^ (7-l)
Now let us insert an expression
into system (6.3). Then we obtain the following relations:
Y=%+PrBl, and
Substituting (7.3) into (7.4) we obtain a nonlinear differential equation
$ = 5”f’(82 + (a2 - 4 P - &P21. (^) (7.5)
If we determine p by Eq. (7.5) and then if we determine y by equation (7.3), the quantity (7.2) is a solution of system (6.3). Equation (7.5) has the following form:
3 = P+l{f(t> + W) P + g(f) P”>s (7.6)
where f, h and g are linear in a, , --*, urn and polynomials in 5-l such that
f(t) = w-9, 45) = %I+ qs-l), m = we') (^) (7.7)
and h, is a nonzero constant. Actually h, is equal to 4. We shall now state a fundamental lemma concerning nonlinear differential equations of type (7.6).
94 HSIEH AND SIBUYA
LEMMA 1. Let f, h and g be polynomials in t-1 whose coeficients are linear in a, , ***, a,. Suppose that
f(O = w-9, WI = ho + O(P), g(5) = 0(5-Y,
where h, is a nonzeyo constant which is independent of a, , *me,a,,,. Then, the differential equation
2 = P+'{f + h(5)p +g(t>p2}
has a unique formal solution
m = i PNPN, N=l
(7.8)
where the quantities pi are polynomials of a, , e*+,a, and independent of E. Let 8 be a suficiently small positive constant. Then there exists a unique solution p(t) of Eq. (7.6) which satisfies the following conditions:
(i) For each positive constant r there exists a positive constant N, such that p(t) is holomorphic with respect to (5, a, , a*., a,) in the domain defined by
ItI >NT> I al I2^ +^ I a2^ I2^ +^ *a-+^ I a, I2^ < r,^ (Ocr<+co),
I arg ho + (m + 2) arg 5 I -C 4 - 6; (^) (7.9)
(ii) P(E) E $(I) u@fmm!y on each compact set in^ the (a, , e-e,a,,,)-space as f tends to infinity in the sector
Iargh,+(m+2)arg5~<~-8. (7.10)
REMARK 6. If 8 tends to 0, N,. may tend to infinity. Hence the domain (7.9) depends on 6 and the solution p also depends on 6. However, because of the uniqueness of p, the solution p defines a unique analytic function. In this sense, p is independent of 6.
- PROOF OF THEOREM 1: PABTIII
In this section, we shall complete the proof of Theorem 1 by the use of Lemma 1. Lemma 1 will be proved in Sections 9-13.
Then
HSIEH AND SIBUYA
is a solution of system (6.3), where the path of integration is taken in the sector (8.3). It is easy to see that the solution (8.6) is holomorphic with respect to ([, a, , ..., a,) in the domain (8.1) and we have
*z Iwo + f w&Nl E(6) N=l
uniformly on each compact set in the (al , *a., a,)-space as [ tends to infinity in any closed sector which is contained in the sector (8.3), where wN are two- dimensional vectors whose elements are polynomials of a, , a,a, and inde- pendent of 5, and 1 wg = [I 0. (8.8) Put
Then U(X) is a solution of system (6.1). If we can prove that U(X) is an entire function of (x, a, , ..., a,), then the proof of Theorem 1 will be completed. To do this, let Q(X) be the two by two matrix such that
qp = A(x) Q(x), W) = 12,
where 1, is the two by two identity matrix. The elements of the matrices Q(X) and @(x)-l are entire functions of (x, a, , sm.,a,), and U(X) must be given by u(x) = O(x) @(x&l 24(x0). (8.10)
Now let (alo, *a., u,O) be fixed, and consider a small neighborhood V of this point. Then we can choose x0 so that (to, a, , **, a,) is in the domain (8.1) for every (ul , -a, a,) in V”, where x0 = 5,“. Hence u(xo) is holomorphic in V. This proves that u(x) is an entire function of (x, ai , *em,a,).
REMARK 7. We shall prove the relation (2.3). The direct computation shows that Lx&y = - (1 + x-“P(x)> - + ??zp-‘, a..#) = {I + Lx-“P(x)> - 8 n@m+2’, /J(E) = (1 - X-“P(x)> + 4 m&-(m+2), p2(.$) = - (1 - .-“P(X)} + + m&-(m+2).
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS 97
Hence
Thus
+) - %(S> = 31 + .-“W>,
/l,(.g p2(f) = (4 my (-2(m+2) - {I - Pqx)}“.
@2(-i) - %(tq2 + 4!%(t) A(5) = 42x-mw + 4 (4 mJ25-2(m+2).
On the other hand, from (7.9, it follows that
34(‘$P(0 = ~2G3 - 43
- (^) [ {~2(t) - 4t>>2 + 41S,(t)B2(0 - 4f-(“+1) MO $I:
Hence
r(5) = %(l3 + p(t) A(5) = - 2 Pq~)}’ + O(Pm+2)).
This implies the relation (2.3).
9. PROOF OF LEMMA 1: PART I
We shall now prove our fundamental lemma in several steps. First, we shall construct the unique formal solution (7.8) of the differential equation (7.6). By assumption
f(E) = i1h5-N A!) =&NCN
where the quantities fN , gN , hN are linear in a, , a,a, , but they are inde- pendent of 5, while h, is a nonzero constant. Then the quantities p, are uniquely determined by
N-l N-l
- h&N = fN + c hN-kpk + c gk.PdPN-k-t k=l k+d=l
(N = 2, 3, *.-, m + 2),
N-l N-l
- hOpN = fN + c hN--KPk + c^ gkj-@N-k-t^ +^ (N^ -^ m^ -^ 2)p,,-, k=l k+d=l (N 3 m + 3). This proved the first part of Lemma 1.
4Q9/WI-
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS 99
where S is a small positive constant and M is a positive constant. Let TI and T, be the tangents of the circle / s 1 = M at s = si and s = sa , respectively. Let T, intersect the line: arg s = (#) n - S at s = si’ and let T2 intersect the line: arg s = S - (8) n at s = sa’. Let us denote by Ys, the domain which contains the sector
/ arg s / > 4 7r - 6, IsI >M’ for a sufficiently large positive constant M’, and which is bounded by the following arcs:
s= -Texp{i(#?r-S)} (^) (- 00 < 7 < - I Sl’ I), s = sl’ + T exp {i arg (s, - si’)} (^) (0 < T < I Sl - 5’ I>, s = MeiT (^) (I 7 I < 77 - ic 9 s = s2 + T exp {i arg (sz’ - sa)) (^) (0 < T < 1 S2- s; I>, s = 7 exp (i(6 - 8 1z)) (^) (I s; 1 < T < + a).
At each point s of 9sM we can construct a straight line
u = S + Teie (^) (0 d T < + 00)
so that the line (10.1) is contained in 9& and that
(10.1)
LEMMA 3. Let positive constants 6 and p be given, where 6 is su@iently small, while p is arbitrary. Then there exist positive constants Me,, and L, such that we have
I
a j u I--P1es-O/ 1da 1 < L, 1s I--P (^) (10.3)
for
s E %Msp 3 (10.4)
wherel, is independent of p, and thepath of integration is the straight line (10.1).
11. PROOFOF LEMMA 1: PART III
As the third step, we shall derive an integral equation from the differential equation (7.6). First of all fix a positive constant Q arbitrarily once and for all. Then consider the sector 9 in the f-plane which is defined by
I arg 4 + (m + 2) arg 4 I < 3 =r, I-51 >Q.
100 HSIEH AND SIBUYA
Let C@rbe the domain in the (at, *a, a&space which is defined by
where I is a positive constant. By the use of Borel-Ritt theorem [1], we can construct a function&Q so that $r is holomorphic with respect to (4, a, , ..a, a,,J in the domain Y x 9,. , and 4, and d$,/df admit the uniformly asymp- totic expansions
for (aI , -.., a,) E CS?as 5 tends to infinity in the sector 9’. Put P = P + 6&O.
Then the differential equation (7.6) is reduced to
3 = Em+‘(/43+ u9 q + b(5)s”},
(11.1)
(11.2)
where
a3 = W) + &?(5)m), %(8 = R(5).
These three quantities are holomorphic in 9 x g,, and we have
P,(4)= 0 (11.3)
and
&M = &J + qY), %(5) = O(P) (11.4)
uniformly for (aI , ..a, urn) E .9? as 6 tends to infinity in 9. The relation (11.3) can be derived from the fact that the asymptotic expansion of j,(f) is a formal solution of the differential equation (7.6). Put 4(‘!) = kl + d,(E).
Then the first relation of (11.4) implies that
cu4) = w-
uniformly for (al , -**, a,) E .9JTas [ tends to infinity in 9’.
(11.5)
102 HSIEH AND SIBUYA
Define the successive approximations in 9& by
Then, by the use of Lemma 3 and (12.2), we can prove that
I Q&3 I d K I E 1-l
in the domain 9s, x 9,.. Now, by the use of (12.3) we can prove the
uniform convergence of {qk(t)} in 5& X 9,. Put
Then q(t, Y) is a holomorphic solution of the integral equation (11.6) such that I 4(5,y) I < K I E 1-l in Sp,, X .C8r+ It is easy to prove that we have
uniformly for (a, , es.,a,) E 9Jr as [ tends to infinity in 9&. Put
Then P = ~(6, Y>^ is a solution^ of the^ differential^ equation^ (7.6)^ such^ that p(f, I) is holomorphic with respect to (5, a, , .*., a,) in Sp,, x 9,. Further- more (12.6) implies that
P(E,4 ei, $(%)
uniformly for (a, , **a,a,) E .9r as 5 tends to infinity in 9&.
13. PROOFOF LEMMA1: PARTV
If we can prove that ~(6, r) is independent of I, then the proof of Lemma 1 will be completed. To do this, let us consider ~(6, YJ and p(f, Y.J, where rl > r2 > 0. They are defined in 9& x^ 9,.1 and Ys,^ x^ Br,^ , respectively. The constants ikli and Ma are determined in the marker described in Sec- tion 12.
SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS 103
Let 9&C 9sM, and YaM,. Then p(t, ri) and p(E, Ye) are defined and satisfy the differential equation (7.6) in 9& x 9,,. Furthermore we have
P(5,4 - PC& 12)z 0
uniformly for (a, , **e,a,)^ E grg^ as 5 tends to infinity^ in 9&.^ Put
u(5) = I@, 5) -I+$, y2>.
Then, in Y8, x .9r, , u satisfies the differential equation
where
J(S) = 45) + g(%)ML 5) + PC59YJ>’
J(t) satisfies an inequality
I .I(0 - 4, I < K I 6 I-’
in y& x DrI , where^ K^ is a constant.^ Let^ & be an arbitrary^ point^ of Sp,,.
Then we have
where
and
5 = h&n + 2)-i p+2, s, = h&n + 2)-l p+‘J
u = h&n + 2)-i 7$+2.
In order that we have u(f) z 0 as E tends to infinity in 9&, we must have u(&,) = 0. Since 4, is arbitrary, we have u(f) = 0 in YSM x .9,.,. This completes the proof of Lemma 1.
hWERENCBS
- K. 0. FRIEDRICHS. Special topics in analysis. Lecture notes, N.Y.U., 1953-4.
- Y. SIBUYA. On the problem of turning points, MRC Technical Summary Report No. 105, December 1959.
- Y. SIBUYA. Simplification of a system of linear ordinary differential equations about a singular point. Funkciuhj Ekvocioj 4 (1962), 29-56.
