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+--- Thread: An intuitive idea log*^[n](F(exp*^[n](z0))) (/showthread.php?tid=1301)
An intuitive idea log*^[n](F(exp*^[n](z0))) - tommy1729 - 03/03/2021
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 ;
log*^[n](F(exp*^[n](z0)))
Looks alot like the 2sinh method and many others.
BUT PERHAPS BETTER BEHAVED ??
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
regards
tommy1729