Yeah, it’s nifty about math that you can come up with the major ideas independently on your own.
I mentioned above that the exponential function can’t be smoothly extended to the point at infinity in the Riemann sphere. I should have also made it clear that it can’t be smoothly extended to the unsigned infinity in the real projective line either (to make it continuous, we would need e^(-infinity) = 0 but also e^infinity = infinity). C’est la vie.