Can exponential functions have "complex" solutions?

Question

Do exponential functions, take $e^x$, have imaginary solutions just as a function like $\sqrt x$ has complex solutions? I know $e^{i\pi} = -1$ but I'm not sure how this works or relates.

Answer 1

If $w\in\mathbb C\setminus\{0\}$, you can write it as $r(\cos\theta+i\sin\theta)$, with $r>0$ and $\theta\in\mathbb R$. Then$$e^{\log r+\theta i}=r(\cos\theta+i\sin\theta)=w.$$