The iteration of holomorphic functions are holomorphic functions.
If a holomorphic function is described by a flow \[f^s(f^t(z))=f^{s+t}(z)\] then it converges where \[s,t\in\mathbb{N}\]. Also since \[f^s\] converges for \[s\in\mathbb{N}\] and the iteration of holomorphic functions are holomorphic functions, then it converges where \[s,t\in\mathbb{Q}\].
Since rational numbers can be taken arbitrarily close to each other while being convergent, an irrational number can only be divergent if the first derivative goes to infinity.
Since a first derivative of a holomorphic function can't go to infinity except at infinity,
\[f^s(f^t(z))=f^{s+t}(z)\implies s,t \in \mathbb{R}\] converges.
If a holomorphic function is described by a flow \[f^s(f^t(z))=f^{s+t}(z)\] then it converges where \[s,t\in\mathbb{N}\]. Also since \[f^s\] converges for \[s\in\mathbb{N}\] and the iteration of holomorphic functions are holomorphic functions, then it converges where \[s,t\in\mathbb{Q}\].
Since rational numbers can be taken arbitrarily close to each other while being convergent, an irrational number can only be divergent if the first derivative goes to infinity.
Since a first derivative of a holomorphic function can't go to infinity except at infinity,
\[f^s(f^t(z))=f^{s+t}(z)\implies s,t \in \mathbb{R}\] converges.
Daniel