I have no problems with this route to show that e^ix = cos(x) + i sin(x) but to do it properly does require showing that this is the unique analytic continuatio

I have no problems with this route to show that e^ix = cos(x) + i sin(x) but to do it properly does require showing that this is the unique analytic continuation of the exponential function. The alternative method is to note that exp() is defined as the unique function satisfying exp(0) = 1 and dexp(ax)/dx = a exp(ax). So exp(ix) describes a point moving orthogonal to it's current position, which means it's a circle, and since it's speed is proportional to it's distance from the origin it is moving at a constant speed. So e^ix is just a point x radions around the unit circle.