This isn't a proof, and while that may be obvious to you, it's not obvious to everyone who sees it. There exists no logical path to the statement from the prev

This isn't a proof, and while that may be obvious to you, it's not obvious to everyone who sees it. There exists no logical path to the statement from the previous definition of e^x because, up to this point, e^x is a function on real numbers. e^ix is nonsense until you define e^x in complex coordinates, at which point the proof needs to rely on properties of that definition. what you have posted misses the nature of the insight. If two analytic functions on the complex numbers agree on uncountably infinitely many points, then they agree everywhere they are defined. This means that there is a unique analytic function that maps (ix)[x \in R] to the circle and (x)[x \in R] to the natural exponential, and they are the same complex function. Without the knowledge that analytic continuations are unique, the statement is entirely (pun intended) meaningless.