Mathematical Analysis by Tom Apostol (section 1.26) defines $e^{ix} \triangleq \cos x + i\sin x$. He then adds an argument for why this definition is reasonable, asserting that this is not a proof.
- Why is this not a proof? Wikipedia seems to think that it counts: https://en.wikipedia.org/wiki/Euler's_formula#Using_polar_coordinates
- Why define $e^z$ in the first place instead of proving it (e.g. using the standard Taylor series proof)?
Here is the relevant section of the text:
You can decide what to define and what to prove. If you define $e^z$ as Apostol does, then you have to prove that it is given by its Taylor series at every point. If you define it to be given by the Taylor series, then you have to prove Euler's formula. When Apostol says that this is not a proof it is because you do not have a previous definition of the complex exponential and, of course, there is nothing you can prove about it. But you can say that he provides a proof of the fact that there is only one such function satisfying the formal differentiation rules for the exponential. If you had defined the exponential as a function satisfying those formal properties, his argument would be a proof of uniqueness of such an object and a proof of the formula.