There’s nothing magical here, the first term describes a 180 degree rotation around a circle of radius 1, landing you at -1. Then you add 1 and you land at the Center of your circle, (0,0).
So why does the first term describe a 180 degree rotation?
People learn that 2^3 means to take three copies of 2 and multiply them together, and that i^2 is -1, so what does it mean to take (i pi) copies of e and multiply them together?
The part you are describing is not where the magic resides ...
