I just read that an elegant proof exists that the law of exponents also holds for complex numbers ($a,b,z$ all complex): $e^{a+b}=e^ae^b,$ which only uses the definition that $y=e^{zt}$ is a solution to $dy/dt=zy,$ with initial condition $y(0)=1$, so in particular $e^z=y(1).$
I can only find a proofs which use the trig-representation of complex numbers.
Can anybody help?
Thank you!