03/14/2021, 05:32 PM
(03/14/2021, 05:00 PM)tommy1729 Wrote: When I came up with the 2sinh method I thought about similar things.
But I believe all iterations of type f(v,z)= 2*sinh^[v] (log^[v](z)) for nonzero v are problematic.
For all positive integer v this seems to be the case anyway. And noninteger seems even worse at first.
Also for v = 1 this is the only rational iteration.
These are still very different from ln^[v](2sinh^[t](exp^[v](z))) or log^[v+a+b](2sinh^[v+c](z)) so not like the 2sinh method or similar by far.
With " problematic " I mean things like divergeance , not analytic and similar problems.
however iterations of g(v,z)= 2*sinh^[v] (h^[v](z)) or the function g(v,z) itself may be interesting.
For instance if h is the logarithm base eta and z is large. That might related to the base change method.
Im not saying that all works fine and easy.
Just a little comment
regards
tommy1729
We could for instance achieve a " hyperbolic base change constant ". If that almost equals the "normal base change constant " then this might be used to show that there is a "problem" ?!
Where problem could be many things like " ill defined " , " just an approximation " or not analytic.
I have not investigated hyperbolic base change constants.
Just one of the possible directions ...
regards
tommy1729