using sinh(x) ?

[tommy1729]

There are many ways to solve f(f(x)) = exp(x) or g(g(x)) = 2sinh(x).

If g < f then g is uniquely so.

Also using the 2sinh method on g gives us f.

NOTICE it is impossible to directly solve for g in the 2sinh method.

So we must use g < f.

This implies a bijection between g and f , by the 2sinh or equivalently the inequality.

Since there exist both analytic half iterates for both 2sinh and exp , for Every f there is a g and vice versa , we get the next BIG conclusion ;

There MUST exist analytic solution of type 2sinh method.

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That is nice.

But one wonders if f and g related as above must

1) both be analytic or both not.

?

In other words is the bijection actually univalent ?

I kinda repeat myself , but I assumed it belongs here better , and it was not understood by most.

Regards

tommy1729