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Full Version: An intuitive idea log*^[n](F(exp*^[n](z0)))
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Thinking about my 2sinh method, I came up with an intuitive iteration that might be interesting.

let z0 = a0 + b0 i  with e < a0 , 0 =< b0 < 1. 

Now define exp*(z) as abs( exp(re(z))*cos(im(z)) ) + abs( exp(re(z))*sin(im(z)) ) i.
notice exp*(positive real) = exp(positive real). 

Now define log* as the appropriate log branches.

Let z_n = z_n(z0) = exp*^[n](z0).

Now consider for various functions F ;


Looks alot like the 2sinh method and many others.


Also how fast does this sequence z_n grow ?

Is log*^[n](F(exp*^[n](z0))) analytic ? equal to the 2sinh method for F an iteration of 2sinh ??

The idea is to avoid the need for analytic continuations and just keep iterating without issues.

Is log*^[n](2sinh(exp*^[n](z0))) = exp(z0) , exp*(z0) or conjugate exp(z0) ??

Another old sketchy idea but worth reconsidering imo