By Geertrui K. Immink (auth.)
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Extra resources for Asymptotics of Analytic Difference Equations
1 2 . Let S be a region of the complex plane such that d ( S , 0 ) > 1. Let ~(~) = ~ - a i ~ I Y + ~ where ~ 6 ~, a E ~ , logI~ I +~(~), ~61R, SES, 7 6 (0,1] and ~ is an analytic function on S with the following properties I~(~)I <_ cI~! I-~ • ~s, l+'(+)f
25)). provided p is suitably chosen (cf. 8) holds requires a slightly more sophisticated definition of the regions S(R), which will be given at the end of this section. First we shall collect some relevant information concerning the shape of the regions x%(Sp(R)) for % > 0 (and, more particularly, for % 6 (0, I])and see what further conditions have to be imposed on p. From now on we drop the subscript p in Sp (R). + By ~ ( 6 ) and ~%(6) we denote the arguments of the left and right derivative of X% 0 o in 6, respectively.
Hence it follows that there is at least one B 6 [~k,~k + ~] with the property + -- that cos(B+arg ~ ) > 0 . Let ~ 6 [~k,~k+~] be determined so that Re(ei~) = max Re(ei8~). k K The preceding argument shows that Re (el~) > O. 52) We now take %(t) to be the half-line from t to infinity, with directional angle ~. 15) that %(t) c Xk(S(R)) for all tEXk(S(R)) and, consequently, that C(s) = S(R) for all s6 S(R). 12. 12 TEXk(S(R)). 53) and applying lemma (cases (i) and (ii)) we obtain the estimate Hillg,a,r+1_k S(R) < K llfll, R>1, provided g