Download e-book for kindle: Asymptotics of Analytic Difference Equations by Geertrui K. Immink (auth.)

By Geertrui K. Immink (auth.)

ISBN-10: 3540138676

ISBN-13: 9783540138679

ISBN-10: 354039060X

ISBN-13: 9783540390602

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Extra resources for Asymptotics of Analytic Difference Equations

Example text

The difference between A and A' lies in the definition of the paths C(s) and C'(s). The first is defined as the directed segment from sh to s' (= s - I ) , whereas C'(s) is the directed segment from the point sh' on the boundary of S'(R) to s'. Now let fEBg,a,r(~(R)). The restrictions of f to S'(R) and S(R'), will also be denoted by f. By Cauchy's theorem we have, for all sES'(R) [~ S(R'), A'f(s)- Af(s) =Yo(S) S Y (~)-If(~)d~, C' (Sh+ ½) o or, equivalently, A'f(s)-Af(s) =Yo(S)Yo(Sh+½) -I I' (Sh+½), where I' is defined analogously to I (cf.

By k I(A) and k2(A) we shall denote the sets k I(A) = {kEk(A) , k~} , and respectively. ,~N_I}. ,N}. Furthermore, we choose a number p such that max k

$ N ' o(~) = {s~+ (s o _ sO ) ~ - ~ h } I / p h+1 h Sh+1-$h if ~E [$h,~+i ], hE {0 .... N - I}, is contained in S*(R). 6) The closed region consisting of all points lying to the left of or on the curve o will be denoted by Sp(R).

I. Re ~I >0" Then we take + ~I =al ' ~ 2 = ~ +~ " (If ~2- ~I < ~ this definition implies that ~I =~I and ~2--~2' cf. 13)). 12) and (iii) , the + interval [~i,~I] contains at most one l-singular direction of A. Let the determinations of Pl and of arg ~I be chosen in such a way that ~ - arg ~I is the one l-singular direction, if there is any, contained in [ ~ , ~ ] , the largest l-singular direction that is smaller than ~ . + ~ + ~I -2 < ~I + arg(~1 - 2hi) < and 3 ~< ~ + ~+arg(~1+2~i)<~+~ 3 ~. or else Consequently we have 45 - + Since [~i,~i] c (0,~) it follows that A Re{e ~(~1+2ei)} > 0 2.

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


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