Where is the flaw in this logic regarding complex numbers?

Question

Since $e^{1+2\pi i n} = e$ for any $n \in \mathbb Z$, one could write $$e^i = \left( e^{1+2\pi in}\right)^{i} =e^{i-2\pi n} = e^i \cdot e^{-2\pi n},$$ which suggests $e^i$ is either equal to zero or multi-valued, whereas Euler's formula and power series suggests that $e^i$ is nonzero and single-valued: $e^i = \cos 1 + i \sin 1$. Obviously, something is wrong here, but what?

Answer 1

When $n=1$, writing $$\left( e^{1+2\pi in}\right)^{i} =e^{i-2\pi n}$$ is what's wrong. It's also wrong for any other non-zero $n$, but I think it's nice to concentrate on a single value where it's false.

You are right that there is a (single valued) function corresponding to $e^z$ - let's call it $\exp(z)$ - that satisfies $\exp(z+w)=\exp(z)\cdot \exp(w)$ for all $z$ and $w$.

There is not a single-valued function $pow(z,w)$ that satisfies

$$pow(pow(z, w_1), w_2))=pow(z, w_1*w_2)$$

for anything other than a highly restricted subset of complex $z,w_1$, and $w_2$. The first "equation" in this answer assumes that there is.

If you're willing to allow multiple valued functions, you could possibly allow more values for the three variables, but I'm not sure about the details.

Additionally, in the comments you have referred to a "proof that $i^i$ is multi-valued". I don't know of such a proof. I do know of arguments motivating the definition of $a^b$ for complex $a$ and $b$, and those define it to be a multi-valued expression, but I don't think I've ever seen any one of them try to make any claims relating ${a^b}^c$ and $a^{bc}$. (Though I could easily be wrong on that last point - it's been a while, and I've stopped trying to connect the two expressions.)