tommy's simple solution ln^[n](2sinh^[n+x](z))

[tommy1729]

Consider f(z,x) = Lim(n --> oo)  ln^[n] ( 2sinh^[n+x] (z) ).

This simple Function satisfies exp(f(z,x)) = f(z,x+1).

So we have a simple superfunction that requires only the real iterations of 2sinh(z).

Notice lim ( n --> oo) 2sinh^[n]( 2^(z-n)) is a superf for 2sinh.

f(z,x) could be analytic for re(z) > 1.

Also , is it really new ?

Or is it the ( analytic continuation ? ) of the 2sinh method ?
It sure is very similar.

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Mick wondered if F^[n] ( g^[n] ) is analytic for f = sqrt and g = x^2 +1.

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Regards

Tommy1729