Exceptional values of solutions of linear differential equations

Mathematische Zeitschrift

Mafia. Z. 201, 317-326 (1989)

9 Springer-Verlag1989

Exceptional Values of Solutions of Linear Differential Equations Norbert Steinmetz Universit/it Karlsruhe, Mathematisches Institut I, Englerstrasse 2, D-7500 Karlsruhe, Federal Republic of Germany

1. Introduction The solutions of linear homogeneous differential equations L w =_ w(,) + p ~ _ l ( z ) w ( , - t ) + ... + p o ( z ) w = 0

(1)

with polynomial coefficients are either polynomials or transcendental entire functions of positive finite order. If Po ~ 0, then every transcendental solution assumes every value c # 0 infinitely often (exponent of convergence = order), while the value c = 0 plays an exceptional role. It m a y happen that zero is an exceptional value in the sense of Picard, Borel or Nevanlinna (for notations of value distribution theory see H a y m a n [5]). In this paper we are concerned with (a) the distribution of zeros of the solutions (b) the density of zeros of individual solutions and (c) the density of zeros of a fundamental set of solutions. (a) It seems to be well k n o w n t h a t ' m o s t ' of the zeros of an arbitrary solution are lying in arbitrarily small sectors around the Stokes' rays arg z = 0 of Eq. (1). However, in [2] Bank has constructed a simple differential equation with the property that, given any ray arg z = ~o, there exists a solution having infinitely m a n y zeros near this ray. In T h e o r e m 1 and Corollary 1 we give a precise statement of what is meant by ' m o s t ' zeros are lying ' n e a r ' the Stokes' rays: The total n u m b e r of zeros in some large disk Izl < r but outside the 'logarithmic strips' Ao: l a r g z - O l < C

log + ]zl izll/~

(2)

around the Stokes' rays arg z = 0 is of order O (log r). (b) F o r an arbitrary transcendental solution of (1) the value zero m a y be a Picard value (only finitely m a n y zeros of w(z)), a Borel value (exponent of convergence < order) or a deficient value in the sense of Nevanlinna (6 (0, w)> 0).

318

N. S t e i n m e t z

A Picard-Borel exceptional value has always maximal deficiency 6(0, w)= 1. In [12] it is asked whether the converse is also true. This will be proved in Sect. 3 (Corollary 2). (c) Any Eq. (1) with constant coefficients has a fundamental set with Picard value zero. If P is a nonlinear polynomial, then (1) is transformed by w = y e P(z) into an equation with nonconstant coefficients having also a fundamental set with Picard value zero. By a theorem of Frank [4] the converse is also true: If an Eq. (1) has a fundamental set with Picard value zero, then (1) may be transformed by w = y e P(z) into an equation with constant coefficients. It is natural to ask whether this remains true if 'Picard value' is replaced by 'Bore1 value' (see [3]). In Sect. 4 we give a complete answer in the case that (1) may have rational coefficients. A necessary and sufficient condition is that the solution space V is a composite of very simple solution spaces V~(1 - 1, while P~(z) = 0 in all other cases. #v+ 1 Let a r g z = 0 be given and let w be an arbitrary nontrivial solution of (1). Then w=cl w~ + c , w ~ in the sense described above. If we consider only those p o l y n o m i a l s belonging to w~o with c~4: 0, there is exactly one with the p r o p e r t y Re(P~(rei*)-Pv(rei*))~ + oo as r--* oo for 0 < ~ o < 0 + a and every P, q: P~ with cu 4=0. Similarly, there is exactly one p o l y n o m i a l P~ with the same p r o p e r t y in the range 0 - a < q~ < 0, T h e n arg z = 0 is called a Stokes' ray if P~ 4:P~, which m e a n s that in 0 < arg z < 0 + 6 and 0 - 3 < arg z < 0 the b e h a v i o u r of w(z) is ruled o by essentially different functions w~o and w~. O u r basic result will be T h e o r e m 1. Let w (z) be a nonrational solution of (1) and let arg z = 0 be an arbitrary direction. Then there is a polynomial P = P~ such that

(10)

log l w (z) l = Re P(z) + O 0og fz I)

as z ~ oo in 0 < arg z < 0 + h, possibly outside an exceptional set consisting of (i) (countably many) disks [ z - z , [ < ]zu[ 1-", where e is positive and the counting function of the sequence (z,) is 0 (log r); log + [z] (ii) a logarithmic semi-strip 0 < arg z - 0 < C - -

izll/p

9

T h e latter occurs only if a r g z = 0 is a Stokes' ray of w(z) (otherwise we m a y set C = 0). An a n a l o g o u s result holds in O - - h < a r g z < O (with the same p o l y n o m i a l if arg z = 0 is not a Stokes' ray, but with a different p o l y n o m i a l in the c o n t r a r y case).

320

N. Steinmetz

Corollary 1. Let w(z) be a transcendental solution of (1) having Stokes'rays arg z = Oi, 1 r I [zl argz. log + [z[ In this case we restrict a r g z to the range C - - < a r g z < h . At any rate we obtain [z[

I

]

w (z) = e P(~) ~ c~ z p~ Q~ (z, log z) + 0 (1z[ - ") LV= 1

as z ~

log + [zl in C - - _ - < a r g z _ - < h

Izl

(20)

( C = O if a r g z = O is not a Stoke's ray), where

# > 0 m a y be prescribed. Finally, Ilogzl d

c,Q~(z, l o g z ) = Q ~ ( l o g z ) + O ( ~ - )

(z--* oo in 0 < arg z < h), where Q~ is a nontrivial polynomial and d is some nonnegative constant. Relabelling a last time if necessary we obtain from (20)

w(zl=e'Zz~ N

(21)

for some e > 0 (co and fl~ real, 1 _ 0 be arbitrary and denote the zeros of 9 in the half-strip Res_>_0,

0_=89189 (# sufficiently large) then follows ( t u ) - ~ (0) = o (t~).

(27)

But this implies that s = 0 is a zero of ~Po of multiplicity > N, which is impossible since ~o is a nontrivial solution of a N-th order linear differential equation with constant coefficients. Applying L e m m a 1 to (21) with NT=e/2, it is easily seen that the term in brackets is b o u n d e d below by const. [z I- ~/z in C log + [z] ~ arg z < h, [z [ > r o, but outside the sets [z[ [log z - s u [ < 89 e - 7Res~ (28) It is clear that each set (28) is contained in the disk

Iz-z.loo, and so from (43) follows

s"(z)-(4z ~-1 +...) s'(z)+(9z2~'-2+...) s(z)=O(1)

as

z--*oe.

Since s is a p o l y n o m i a l this is only possible if 2 = 1, but this already implies that p and q are constants (see [-13]).

References 1. Ahlfors, L.V. : Complex analysis. New York: McGraw-Hill 1979 2. Bank, S.B.: A note on the location of complex zeros of solutions of linear differential equations. To appear 3. Bank, S.B., Frank, G., Laine, I.: ~ b e r die Nullstellen von LSsungen linearer Differentialgleichungen. Math. Z. 183, 355-364 (1983) 4. Frank, G.: Picardsche Ausnahmewerte bei L6sungen linearer Differentialgleichungen. Manuser. Math. 2, 181-190 (1970) 5. Frank, G., Wittich, H.: Zur Theorie linearer Differentialgleichungen im Komplexen. Math. Z. 130, 355-364 (1973) 6. Hayman, W.K.: Meromorphic functions. Oxford: Clarendon Press 1964 7. Hille, E.: Ordinary differential equations in the complex domain. New York: Wiley 1976 8. Schwengeler, E.: Geometrisches fiber die Verteilung der Nullstellen spezieller ganzer Funktionen. Dissertation, Zfirich, 1925 9. Sternberg, W.: Ober die asymptotische Integration von Differentialgleiehungen. Math. Ann. 81, 119-186 (1920) 10. Valiron, G.: Integral functions. New York: Chelsea 1949 11. Wasow, W.: Asymptotic expansions for ordinary differential equations. New York: Wiley 1965 12. Wittich, H.: Defektfreie LSsungen linearer Differentialgleichungen. Arch. Math. 7, 459-464 (1956) 13. Wittich, H.: Zur Kennzeichnung linearer Differentialgleichungen mit konstanten Koeffizienten. Festband zum 70. Geburtstag yon Rolf Nevanlinna (pp. 128-134). Berlin Heidelberg New York: Springer 1966

Received June 6, 1988