Comment by dboreham

IANAM but I'd go with "it's implicit in how complex numbers are defined". Complex numbers are a thing made up by humans (as are negative numbers), and we got to define i as "up the y-axis". Once you do that, and note that a rotation is therefore cos angle plus i sin angle, add in that e^something is an eigenfuncion of differentiation, and you're pretty much there.

Fwiw I think it's Maclaurin series for this.

Edit: obviously should be j not i.