Could someone please explain to me what the following Big-O notations mean?
What the author mean when he says that $|\phi(s)|=O(e^{|\tau|^\delta})(|\tau|\rightarrow \infty)$ uniformly on F?
Does it mean that $\exists M>0, \tau_0>0$ s.t $\forall \tau>\tau_0$ and $\forall s\in F$ s.t $Im(s)>\tau_0$ $|\phi(s)|\leq Me^{|\tau|^\delta}$
or is it $\exists M>0, \tau_0>0$ s.t $\forall \tau>\tau_0$ and $\forall s\in F$ $|\phi(s)|\leq Me^{|\tau|^\delta}$
I am just confused with the uniformly part.
I have a similar issue with the statement below:
Can someone please provide a rigorous definition of the O and ~ notations above?