07/28/2010, 02:53 PM
as said before exp exp exp ... (z) does *not converge* in a neighbourhood of z when im(z) =/= 0.
i believe my algoritm is coo on the reals and by analytic continuation / mittag leffler expansion we can construct " the " (imho i.e. "my") sexp.
by * not converge* i mean it is chaotic , since of course exp exp ... (2) also diverges.
this is so because e^(x + yi) = e^x ( cos y + sin y i ) and large y can become arbitrary close to a multiple of 2pi.
although analytic continuation solves the problem - assuming it works and it has period 2pi i - i would like to investigate further the behaviour of the exp iteration.
for instance , is it true that the neighbourhood of z always contains exactly one non-chaotic value ? call it hp.
assuming we can find hp for every z as a limiting sequence going to zero in the correct way.
is it then possible to extend my formula to :
exp^[z1,z] = ln ln ln ... 2sinh^[z] exp exp exp ... ( z1 + hp(z1) )
regards
tommy1729
i believe my algoritm is coo on the reals and by analytic continuation / mittag leffler expansion we can construct " the " (imho i.e. "my") sexp.
by * not converge* i mean it is chaotic , since of course exp exp ... (2) also diverges.
this is so because e^(x + yi) = e^x ( cos y + sin y i ) and large y can become arbitrary close to a multiple of 2pi.
although analytic continuation solves the problem - assuming it works and it has period 2pi i - i would like to investigate further the behaviour of the exp iteration.
for instance , is it true that the neighbourhood of z always contains exactly one non-chaotic value ? call it hp.
assuming we can find hp for every z as a limiting sequence going to zero in the correct way.
is it then possible to extend my formula to :
exp^[z1,z] = ln ln ln ... 2sinh^[z] exp exp exp ... ( z1 + hp(z1) )
regards
tommy1729