06/28/2010, 11:18 PM
im going to ask some questions about sexp , and i mean all proposed solutions of sexp.
(question 1)
does the following hold :
d f^n / d x^n sexp(slog(x) + k) > 0 for all positive integer n and all positive real k ?
(question 2)
since slog(z) (base e) has period 2pi i why doesnt sexp(z) look like a log spiral ?
or does it , like having a branch cut at real x < -2 ?
(question 3)
what happens to limit cycles and n-ary fixpoints ??
sure we can set the fixpoints exp(L) = L at oo i but how about the fixpoints of exp(exp(.. q)) = q and limit cycles of the exp iterations ...
e.g. let e^q1 = q2 , e^q2 = q1 , if we want half-iterates , 1/3 iterates and sqrt(2) iterates to have the same fixpoints , this is a problem , not ?
perhaps we can 'hide' L at +/- oo i ( like kouznetsov ) and 'hide' the other points at - oo ??
(question 4)
do all 'analytic in the neigbourhood of the positive reals' sexp have the fixpoints exp(L) = L at oo i ?
( i know that [L,sexp(slog(L)+o(1))] cannot be in the analytic zone , but maybe [L,sexp(slog(L)+o(1))] isnt part of the analytic zone )
(question 5)
slog(z) base x is not holomorphic in x in a domain containing the interval [a,b] if eta is between a and b.
why is that ? i know that the real fixpoint dissappears but still ...
sorry if those are FAQ or trivial Q.
regards
tommy1729
(question 1)
does the following hold :
d f^n / d x^n sexp(slog(x) + k) > 0 for all positive integer n and all positive real k ?
(question 2)
since slog(z) (base e) has period 2pi i why doesnt sexp(z) look like a log spiral ?
or does it , like having a branch cut at real x < -2 ?
(question 3)
what happens to limit cycles and n-ary fixpoints ??
sure we can set the fixpoints exp(L) = L at oo i but how about the fixpoints of exp(exp(.. q)) = q and limit cycles of the exp iterations ...
e.g. let e^q1 = q2 , e^q2 = q1 , if we want half-iterates , 1/3 iterates and sqrt(2) iterates to have the same fixpoints , this is a problem , not ?
perhaps we can 'hide' L at +/- oo i ( like kouznetsov ) and 'hide' the other points at - oo ??
(question 4)
do all 'analytic in the neigbourhood of the positive reals' sexp have the fixpoints exp(L) = L at oo i ?
( i know that [L,sexp(slog(L)+o(1))] cannot be in the analytic zone , but maybe [L,sexp(slog(L)+o(1))] isnt part of the analytic zone )
(question 5)
slog(z) base x is not holomorphic in x in a domain containing the interval [a,b] if eta is between a and b.
why is that ? i know that the real fixpoint dissappears but still ...
sorry if those are FAQ or trivial Q.
regards
tommy1729
