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 [exercise] fractional iteration of f(z)= 2*sinh (log(z)) ? tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 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 « Next Oldest | Next Newest »

 Messages In This Thread [exercise] fractional iteration of f(z)= 2*sinh (log(z)) ? - by Gottfried - 03/12/2021, 12:53 PM RE: [exercise] fractional iteration of f(z)= 2*sinh (log(z)) ? - by JmsNxn - 03/12/2021, 06:19 PM RE: [exercise] fractional iteration of f(z)= 2*sinh (log(z)) ? - by JmsNxn - 03/13/2021, 11:27 PM RE: [exercise] fractional iteration of f(z)= 2*sinh (log(z)) ? - by tommy1729 - 03/14/2021, 05:00 PM RE: [exercise] fractional iteration of f(z)= 2*sinh (log(z)) ? - by tommy1729 - 03/14/2021, 05:32 PM

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