We all know that an equation of degree $n\in\Bbb N$ has exactly $n$ complex roots, even if we don't know how to find them in closed form. But what do we know about equation with complex degree ? I tried a few ones and I got very different result:
$$z^i=1$$
Give an infinitude of solution of the form: $z=e^{2\pi k}$ with $k\in \Bbb Z$. On the other hand:
$$z^i=z$$
Seems to have only one solution: $z=1$.
I think the main issue is that complex exponentiation is a multivaluated function and so it is complex logarithm as it's inverse. But are there general methods with one can prove that "this equation has only one solution" or "this equation has an infinite number of solutions" and so on ?
