Again on Clausen paradox

Question

I am somehow dissatisfied with the explanations I've found for the following paradox:

$$\tag1 e^{1+2\pi in}=ee^{2\pi in}=e$$ raising both sides to $1+2\pi i$ we have $$\tag2 (e^{1+2\pi in})^{1+2\pi in}=e$$ then expanding the exponent $$\tag3 e^{1+4\pi in-4\pi^2n^2}=e$$

$$\tag4 e^1e^{4\pi in}e^{-4\pi^2n^2}=e$$

$$\tag5 e^{-4\pi^2n^2}=1$$ which is paradoxical as $e \neq 1$. As the Wikipedia article says, the problem is in passages 2 and 3, as the way in which a complex power is defined is $w^z=e^{z\log w}$ where the complex logarithm is multivalued, so $w^z$ is multivalued as well. But the reason why the passage 2, 3 is not justified is not made explicit, and I am trying to do that, rigorously, here.

Starting from point 2, we say that the formula $(e^{1+2\pi i})^{1+2\pi i}$ is in fact a multivalued expression that can be expressed as $$e^{(1+2\pi in)\log(e^{1+2\pi in})}=e^{(1+2\pi in)(1+2\pi in+2k\pi i)}$$ then the value $e^{1+4\pi in-4\pi^2n^2} \neq e$ is obtained when $k=0$, while $e$ is obtained with $k=-n$. So 2 and 3 are invalid because they assume different values of $k$. Is this correct?

Answer 1

I've also just read Wikipedia and a nice explanation here how the logarithm could be clearly defined as a normal unique function: Are my exceptions for exponentiation rules complete? Multi-valued functions are not needed.

I think the key point is that the power function and the exponentiation are different concepts and the "paradox" secretly mixes both.

Answer 2

We may define

$\exp(z)=\sum\limits_{k=0}^\infty\dfrac{z^k}{k!},$

which is by construction an entire function. Thus matches

$e^x=[\exp(1)]^x$

when $x$ is real, but then $\exp(z)$ rendered by this Taylor series does not have the branching that $e^z$ would have in the complex plane. In working with complex variables I always distinguish $\exp(z)$ from $e^z$.