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Download Asymptotics of Analytic Difference Equations by Geertrui K. Immink (auth.) PDF

By Geertrui K. Immink (auth.)

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48) [Yo(S')Yo(~)-If(~)d~l. 39) II(s)l< K f C(s) Let k = d e g r ~ (cf. §2 for the definitions of q and degr). This implies that k6k(A). We shall distinguish the following cases. Case I: k ~ 0 and (g,lal) < (k,c). 2: g = k = 0 , r < r o. 2: g = k = 0 , r>r -- . O Here c,b,b' and r are real numbers, to be specified later. Note that case I o relates to values of k such that k > k i , whereas cases ~ a n d I[I relate to values for which k < k i _ I . From now on we put ~k = ~. Io k+0, ~,lal) < (k,c).

78) s-h r ~< (sin ~)r, < (h+ I) r, that s6 S, h 6 I~. 79) sE S, h 6 ~. and s E S(I), we find sES(1), hEl~. 80) sES(1), h61~. s- 11 ..... Is-hl > h! 74) and the estimates found above it follows immediately that A is an inverse of A B. Observe that the definition of A is independent of the values of a I and a2 and the shape of the regions S(R). 5. Conclusions. 4 we can collect the following information concerning the constants c i and b i. If i ~ g, we may take for c i any positive number smaller than the minimum of all constants c (cf.

E. kCk1(A)). The following two situations may arise. -- (i) There is an ~ C [~k,~k+~] such that Re(e is B) >0. In this case the paths C(s) are defined in the same manner as previously and analogous results are thus obtained. In particular, the mapping A is again an inverse of the difference operator corresponding to B. + -- -- + (ii) Re(eZra) < 0 for all ~ E [~k,~k+~]. 54) for a suitable determination of arg ~. Since ~ ( ~ ) is a continuous function of $, monotonically increasing from k~ 2 - ~ to k~ I (cf.

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