Power with complex exponents

Question

I want to demonstrate that $i ^ i = \mp 1$

$i^i=i^i$

$(i^i)^i=i^{-1}$

$(i^i)^i = -i$

$((i^i)^i)^i=(-i)^i$

$i^{-i}=(-1)^i \cdot i^i$

$\frac{1}{i^i}=(i^2)^i \cdot i^i$

$1=i^{2i} \cdot i^i \cdot i^i$

$1=i^{4i}$

$i^i = \mp 1$

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

What is wrong?

Answer 1

A simplified version of your dilemma is that $$(i^i)^4 \neq (i^4)^i \iff e^{-2\pi + 8k\pi} \neq 1$$ even though they both "look" like $i^{4i}$. Exponent laws are reserved for real quantities: http://mathworld.wolfram.com/ExponentLaws.html

Answer 2

BEFORE you start demonstrating a theorem about $\,\imath^\imath,\,$ you have to DEFINE it.

What is your definition of $\,\imath^\imath\,?$