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+--- Thread: about kouznetsov again (/showthread.php?tid=492)
about kouznetsov again - tommy1729 - 08/12/2010
about kouznetsov's tetration.
lim b -> oo f(a + bi) = fixpoint exp.
that seems true for integer a and integer b but is that true for all real a and b ?
it was said to be periodic , so i guess not.
but is it bounded ? it seems like a double periodic limit at oo then.
in general double periodic is unbounded in its period ...
but this is a special case ; approaching at oo.
the cauchy contour would work better if we truely had :
lim b -> oo f(a + bi) = fixpoint exp.
that would require that f(x + i) = g(f(x)) and g(x) has fix exp as a strong and unique fixpoint ( i think ).
in that case the cauchy contour should work better ! not ?
but how to construct such a solution ?
this related to another recent post i made : " twice a superfunction ".
regards
tommy1729
RE: about kouznetsov again - BenStandeven - 08/13/2010
(08/12/2010, 12:26 PM)tommy1729 Wrote: about kouznetsov's tetration.
lim b -> oo f(a + bi) = fixpoint exp.
that seems true for integer a and integer b but is that true for all real a and b ?
it was said to be periodic , so i guess not.
It's true for all real a. b isn't free in that formula; the limit does not depend on how b approaches oo. Of course, this only works for bases greater than eta; below that the tetration function would be periodic, as you point out.