Comments on proofs of Euler's identity

You can only proof this identity after you clearly defined what e^(ix) means.

In this sense arguments using calculus and using differential equations are incomplete, without at least a clear definition any argument becomes an heuristic argument.

I know of three ways of defining e^(ix).

(1) From the complex series of e^(z) defined as sum of z^n/n!

(2) as the limit of n to infinity of (1 + ix/n)^n

(3) or directly as cos(x) + i sin(x)

Either way only after defining e^(ix) is that you should show that it has the properties like e^(ia)*e^(ib)=e^(i(a+b)), or (e^ix)'=ie^(ix) that you would expect it to have, and some use them fearlessly.

Number (1) is the most common one, I think number (2) would be a nice thing to use since it is analogous to the real case and can also be interpreted geometrically (as Richard Feynman does for a reference). But using (3) is totally misleading because it doesn't show why it should be true.

That is why I think heuristic arguments are needed to provide "a reason" for us to believe that such a thing should be true. Using circular motion as in my next post seems to me much more simple and much less "out of the blue". While the proof by itself is heuristic it can be made rigorous.