Why do $e^i$ and $i^e$ both have absolute value 1

Question

Why do $e^i$ and $i^e$ both have absolute value 1? I don't know how to view $i^e$ as a complex number. What is its real part and imaginary part?

Answer 1

We define $z^w$ as $z^w=e^{w\log(z)}$ where $\log(z)$ is the multivalued function

$$\log(z)=\text{Log}(|z|)+i\arg(z)$$

where $\arg(z)$ is the multivalued argument of $z$.

Here, we have $i^e=e^{e\log(i)}$. Inasmuch as $\text{Log}(|i|)=\text{Log}(1)=0$ and $\arg(i) =\frac\pi2+2n\pi$, $n\in \mathbb{Z}$, we have

$$\begin{align} i^e&=e^{e\log(i)}\\\\ &=e^{e\left(\text{Log}(|i|)+i\arg(i)\right)}\\\\ &=e^{e\left(0+i\left(\frac\pi2+2n\pi\right)\right)}\\\\&=e^{ie(\pi/2+2n\pi)} \end{align}$$

which clearly has unit magnitude.