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[exercise] fractional iteration of f(z)= 2*sinh (log(z)) ?
#5
(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 Smile

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
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RE: [exercise] fractional iteration of f(z)= 2*sinh (log(z)) ? - by tommy1729 - 03/14/2021, 05:32 PM

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