Further evidence that e^{ix} = cos(x) + i sin(x) is natural is that it fits in nicely with power series representations. If you define cos(x), sin(x), and e^

Further evidence that e^{ix} = cos(x) + i sin(x) is natural is that it fits in nicely with power series representations. If you define cos(x), sin(x), and e^x by their power series centered at 0 then it's straightforward to see that substituting ix into the power series for e^x yields the sum of the power series for cos(x) and i*sin(x) (as long as you accept that theorems about absolute convergence and rearranging terms extend to the complex numbers).