In the book of Complex Analysis by Conway, on page $32$, it is given that
Consider the series $\displaystyle \sum _{n=0}^{\infty }\:\frac{z^n}{n!}$ ;we have that this series has radius of convergence $\infty$. Hence it converges at every complex number and the convergence is uniform on each compact subset of $\mathbb{C}$. Maintaining a parallel with calculus, we designate this series by $$e^z =\exp z= \sum_n^\infty \frac{z^n}{n!},$$ the exponential series of function.
However, the author states the fact $$e^z = \sum_n^\infty \frac{z^n}{n!}$$ as if is it a definition or just a notation, but $e^z$ has a meaning prior to the above statement, so shouldn't the author show the equality of LHS to RHS in here ?
If so, how can we show this ?
It is simply the definition of $e^z$, the complex exponential. What the author did before is showed that the series $\sum_{n=0}^\infty \frac{z^n}{n!}$ converge for any number $z\in \mathbb{C}$. If that wasn't the case then the definition of $e^z$ would make no sense.