Good treatment of Euler’s formula

Question

I need help because I'm a bit confused about the correct way to treat Euler's formula.

If I ask Wolfram Alpha (WA) about $(i^i)^4$ the answer is $e^{(-2 \pi)}$, therefore, I thought, ok this must be equal to $(e^{i 2 \pi})^i$ but, this is not true, WA says that is equal 1...but, it says that its general form is $e^{-2\pi n}$ but that can not be ever 1...so my head explodes here.

Note: I will appreciate a link to a good explanation about that kind of things.

Thank you

Answer 1

The result is obtained from

$$i=e^{\frac{i\pi}{2}+2 k \pi i}$$

then

$$i^i=e^{-\frac{\pi}{2}-2k \pi}$$

and

$$(i^i)^4=e^{-2\pi-8k \pi}$$

Answer 2

The complex exponential does not satisfy $a^{bc}=(a^b)^c$.

Of course the question is how you define the complex exponential in the first place. In gimusi's answer, it is a multi-valued function. Personally, I prefer to define it as single-valued (which means it is not defined or discontinuous somewhere).

Some information is on Wikipedia at https://en.wikipedia.org/wiki/Exponentiation#Powers_of_complex_numbers

More application-oriented textbooks (such as Wunsch, Complex Variables with Applications or Brown-Churchill, Complex Variables and Applications) define $a^b$ as multi-valued. More theory-oriented textbooks (Lang, Complex Analysis or Stein-Shakarchi, Complex Analysis) tend to make $a^b$ single-valued, but only defined on a subset of the complex numbers, and dependent on a choice of a branch of the logarithm.