Multiple exp^[1/2](z) by same sexp ?

[sheldonison]

....
So exp*^[1/2] is not entire because sexp*(z) is not entire.
sexp*(z) is not entire => sexp*(slog*(z)+1/2) is not entire.

So for the same reasons exp^[1/2](z) is not entire , exp*^[1/2](z) is not entire.

we know sexp*(z) is not entire because it has to have log branches or algebraic branches ( coming from sexp*(z-1) = log*(sexp*(z)) ).

regards

tommy1729

What about f(z)=sexp(2*z)? We can generate the function
\( g(z) = f(f^{-1}(z)+1/2)=\exp(z) \), which is entire... I guess that's not fair since the multiple branches as you circle the fixed point of L all agree at integer values of n.

But what if the fixed point is real? Can we say that \( g(z) = f(f^{-1}(z)+1/2)=\exp(z) \) is never entire?