On the Asymptotic Behavior of the Coefficients of Asymptotic Power Series and Its Relevance to Stokes Phenomena

. This paper discusses the relevance of the asymptotic behavior ofthe coefficients of asymptotic power series for the study of Stokes phenomena. By way of illustration a connection problem is considered in the theory of linear difference equations.

This implies that y-ch is holomorphic at and, consequently, Yn__o a,,z-" is a convergent power series.From (0.2) and (0.3) it now follows that c -2 7ri lim Y" (n-1)!" Apparently, the constant c, which plays a central role in the Stokes phenomenon occurring in this example, is intimately related to the asymptotic behavior of the coefficients y,.It is this relationship that forms the subject of this paper.
We shall consider the following situation.Suppose we are given a number of sectors S, v {1,..., N}, which cover a neighborhood of o and a corresponding number of functions y with the following propeies" y is analytic in S and represented asymptotically by a series of the form ,o f,z-" (independent of v) as z , z S, {1,..., N}.Moreover, assume that (0.4) y+l(z)-y(z) c;;(z), z S S+1, v {1,..., N}, j=l where SN+I e2=is1, YN+I(Z) Yl( z e-=), c C, and the belong to a ceain class of analytic functions.We shall establish a relation between the complex numbers cf and the asymptotic behavior of , for n .In some applications this relation may be exploited to "compute" at least pan of the numbers c; from the coefficients , (cf. [9] and Remark 2 herein).
If the y represent (sectorial models of) a resurgent function, our results could be derived from the work of Ecalle (cf.[4]).For the present purpose, however, this assumption is not needed and we shall establish the relation mentioned above in a more direct manner.
The argument is essentially the same as the one we used in [9].It is based on the Propositions 1.1-1.3herein.Proposition 1.1 concerns the propeies of Cauchy-Heine transforms of functions like the ; in (0.4).Proposition 1.2 enables us to construct, from the Cauchy-Heine transforms of the ;, analytic functions H with the same propeies as the y and only differing from they, by a convergent power series in 1/z.The coefficients of the asymptotic expansion H of the H are given by the expression H, =-2i =1=1 c j(t)t "-dt, yc S.
Under ceain conditions, like those mentioned in Proposition 1.3, the saddle-point method may be applied to the integral f ;(t)t "-at to obtain its asymptotic behavior for n-.Themainr esult is stated in Theorem 1. 4.   In 2 this result is applied to a connection problem in the theory of homogeneous linear difference equations. 1.The general argument.Let C denote the Riemann surface of log z.Let zo C, a, fl e N, a < ft.By S(a, fl) we denote the sector S(,)={zeC" a <arg z<} and by S(zo, a, ) the set (1.1) S(zo, a, ) {z e C" a < arg (z-Zo) < , Izl > Izol}, This will also be called a sector.Downloaded 12/18/18 to 129.125.148.19.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpIf S is a sector of the form S S(zo, a,/3), then _S will denote the sector S(zo, a, fl +2r).
Let / Y=o h,z-n be a formal power series in z -1, S a sector of the type (1.1), and h a function on S. We say that h is represented asymptotically by h as z oo in S, and write h(z)'-Y.h.z-", z-ooinS n=O if, for every S' (C) S and every N E N, Any function q which is analytic in a sector S and represented asymptotically by zero (i.e., the series with coefficients equal to zero) as z-o in S, may be written as the difference of two determinations of its Cauchy-Heine transform.The following proposition, due to Ramis, is concerned with the asymptotic properties of this Cauchy-Heine transform.PROPOSITION 1.1 (cf.12, Prop.4.2]).Let o and fl be real numbers such that a < fl, Zo S(a, fl), and let # be an analytic function on S S(zo, a, fl ).Suppose there exist positive numbers M,, n N, such that where y is a half line in S from Zo to with direction (R), has the following properties" (i) h can be continued analytically to , (ii) h(z)-h(z e2i) (z) for all z S, (iii) h is represented asymptotically by z--, as z in .M oreover, for every S' there exists a positive constant Cs, such that sup [z"R.(h;z)[ Cs,mn+l, n .

zS'
Proof Let us suppose that S is a convex set, i.e., /3-7r/2 <arg Zo< a + 7r/2.In that case every half line from Zo to oe with direction O e (a,/3) lies in S. If y has direction O, h is obviously analytic in S(zo, O, (R)+27r).there is a number e(0, r/2) such that a + e < arg (z Zo) </3 + 2 zr e for all z S'.Let z S' and choose (R) E (a,/3) in such  + e < arg (z-Zo)< O + 27r-e.Let Yo be the half line from Zo to with direction (R).For all " Yo the following inequality holds: (1.3) I" zl > z Zol sine > zl \(1 Zo s ) sin e.
If S is not convex the above argument must be adapted in an obvious manner.
PROPOSITION 1.2 (cf.[10], [12]).Let N. Let a, , u{1,...,N}, be real numbers such that a <= O+ < fly +1 if < N and a N aN+ a + 2 < fin ,+1 fll+2.Let z S(a+l, fl) and S S(z, a+l, fl), 1,.., N. Suppose that, for every { 1, , N}, we are given an analytic function on S with the property that ( z) 0 as z in S .L et where y is a half line in S from z to with direction 0 and let and HN+,(z)= h(z).
=1 =1 e functions H have the following properties" (i) For every u {1,., N+ 1} there exists a S(a, ) such that Z+l 1 e 2i and H is analytic on S S( , a, ).
Hence the function h--H1-H1 can be continued analytically to a reduced neighbor- hood of .Furthermore, property (iii) implies that h admits an asymptotic power series expansion in z -1 as z--> in a neighborhood of o and, consequently, h is analytic in a full neighborhood of The next proposition concerns the asymptotic behavior of integrals of the type ,(z)z" dz, ,/ where 3' is a half line and q is an analytic function with the property that q(z)---0 as zoo in some sector S containing 3'.The conditions (iii)-(v) below are purely technical and have been chosen in such a way that the result follows by a straightforward application of the saddle-point method.They might be relaxed or replaced by other conditions.We have merely tried to define a class of functions for which this method works.
PROPOSITION 1.3 (cf.[2, Thm. 7, Remark 6]).Let a and fl be real numbers such that a < fl, Zo S(a, fl ), and let q, be an analytic function on S S(zo, a, fl with the property that (i) exp p(z)-O as z->o in S.
Let g S x -> C be defined by g(z, n)= O(z)+ n log z.Downloaded 12/18/18 to 129.125.148.19.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpSuppose there exists no t such that for all n >= no the following conditions hold: (ii) The equation Og/Oz =0 has a solution sn S such that the half line ynfrom Zo to o through s is contained in S.Moreover, s, as n o. (iii) There exists a number 6) (0, /2) such that and, furthermore, s, (iv) ere exist positive numbers eo and K such that zg(z,n) g(z,n) g Izs,[ < ols, I.
Furthermore, let f be a bounded analytic function on S and suppose there exists a positive number e such that (vi) SUpz.)lf(z)-l[O if n, where I,(e) denotes the segment between s,(1 e e '-) and s,(1 + e e-).

02
Since this is true for every sufficiently small e the result follows.
Suppose that for every u {1,..., N} there exists a sector S S+, a positive integer re(u), and, for every j {1,. . re(u)}, analytic functions f and ff on , satisfying the conditions of Proposition 1.3, and a complex number c such that m() (z) E c;f;() exp 6y(z), s.
u=l j=l Proof There exists z E S S+1 such that S S+1 contains the sector S S(z, a+l, fl).As y and Y+I admit the same asymptotic expansion, it follows that q(z)=y+l(z)-y(z)--.O asz-*inS , uE{1,...,N}.
Obviously, the functions y possess the properties (i)-(iii) mentioned in Proposition 1.2.
According to Proposition 1.2 there exists a function h, holomorphic at , such that y h + H for all v E {1,..., N}.Let Y,--o h,z-" be the power series expansion of h.With (1.4) we find f, h, where y is a half line g, uE{1,..., N}.
The proof is completed by application of Proposition 1.3 to each term of the sum in the right-hand side of the above identity.
Remark 1.If the y as well as the functions ff exp qf are solutions of some homogeneous linear functional equation, the numbers c play a role similar to the Stokes multipliers in the theory of linear differential equations.
Remark 2. If one of the functions M in (1.11) dominates the rest for n -, the corresponding coefficient c may be determined from the asymptotic behavior of )3,.
Remark 3. Propositions 1.1 and 1.2 may also be used to obtain estimates of the growth of the remainder terms R,(y ;z) as n .This will be illustrated by the application to linear difference equations in the next section.
Example.The nonlinear differential equation ( Let denote one of the formal solutions and let S be a sector of aperture less than r.It is a well-known fact that there exists a solution of (1.12), analytic in S and Downloaded 12/18/18 to 129.125.148.19.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phprepresented asymptotically by 33 as z-in $, uniformly on S (cf. [13]).Suppose that Yl and Y2 are two solutions with these properties.Obviously, d b (1.15) d---(Yl-Y2) Yl Yz +-(YI + YlY2 + y22)(Yl-Yz).
Let )--"n___l )g and suppose the coefficients 33 satisfy (1.13).Then we have 3 where h is a bounded analytic function on S, admitting an asymptotic expansion as zo in S. Inserting (1.16) into (1.15)we obtain d---(yl-y)= -2+zh(z) (Yl-Y2) and this implies that Yl--Y2--ce-ZZg3( 1-+ 0 ()) z in S, where c is a complex number.Hence it follows that (1.12) has a unique solution y-, analytic in a left half plane and represented asymptotically by the series Y,---1 33z-" as z in this half plane.Moreover, it is easily seen that ymay be continued analytically to a sector of the form S(Zl, -37r/2, 37r/2), with Zl e C, without a change in asymptotic behavior.
In the first six cases, Og/Oz =0 has a unique solution s, given by n+p (2.19) Let S' S. In each of the cases 1-6 there exists a positive number 6 such that cos (arg z + arg/x) < -7--7, for all z S'.
This implies that, for all z S', Re g(z, n) = -6lzl + (n + Re p) log Izl-Im p arg z.
Hence we easily deduce the existence of positive constants As, and Cs, such that (2.23) sup lexp g(z, n)] < Cs,As,n . zS' Now consider the cases 7-10.There d # 0 and the saddle point sn is a solution of the equation (2.24)   s,, log s,, +-+ 1 d Let h be the inverse of the function z-)z log z (cf.[9, Ex.III], [4, 3.6]).(2.30) s.OZ 2 We easily verify that (2.31) 2(n+p)-dz n+p-dz and the expression on the right-hand side is obviously uniformly bounded on the half plane -d Re z > 0 and thus on S, provided n _-> no, where no is some sufficiently large number.
Let $' S. In each of the cases considered this implies the existence of a positive number 6 such that d cos arg z < -6 for all z S'.
Let 0 < e < 6.Then there exists a positive constant C such that leap g(z, n)l < C exp (-lzl log Izl)lzi", z Downloaded 12/18/18 to 129.125.148.19.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php The expression to the right of the inequality sign attains its maximum as Iz[ h(ne/e)/e and the maximum value is equal to g" S(:,( )r+6, v-1 ) if u { Next, we consider the function f defined by (2.16).The asymptotic properties of h' imply that (2.36) lim h(z)= 1 uniformly on S.
Furthermore, from (2.14) and the fact that p is analytic on either a lower or an upper half plane it follows that (2.37) Ip(z) exp (-2nTriz)-cl <-_ K exp (-27rllm z[), z where K is a positive constant.From (2.36) and (2.37) it is obvious that fff is bounded on S .Moreover, with the aid of (2.20) it is easily seen that, in the case that dij =0, f satisfies condition (vi) of Proposition 1.3.Now suppose that ,6 {1, 2, 4, 5} and d 0 0. Formulas (2.4) and (2.28) imply that IIm s,l as n , where s, denotes the saddle point of g(z, n).With (2.36) and (2.37) it follows that, also in this case, condition (vi) of Proposition 1.3 is fulfilled.
With the aid of Propositions 1.1 and 1.2 we are able to estimate the growth of the remainder terms R (h z) for n , j {1,.., m}.Let { 1,., 6}. S S+ is a sector of the form S(z, , fl).We begin by considering the functions h defined by fr () d, i,j{1 m}, {1,... where y is a half line in S S+ from z to and is defined by (2.17 Proof The first two statements follow immediately from Proposition 1.1 and the propenies of .Now let S' S S+.We can choose a sector S" S S+, of the form S"= S({, &, ) such that S' $".Let be a half line in S" from { to and 2i if(if-z) As h-h is holomorphic at , it is obviously sufficient to prove (2.39) for h o instead of h.Using (2.18), (2.23), and (2.32) and noting that, due to (2.16), (2.36), and (2.37),The result now follows by application of Proposition 1.1.