= Complex exponentiation is not associative
{title2=$(i^i)^i\ne i^{(i^i)}$}
Using principal <complex logarithms>, the inner power is $i^i=e^{-\pi/2}$. Consequently
$$
(i^i)^i=-i,\qquad i^{(i^i)}=\exp\left(i\frac\pi2e^{-\pi/2}\right),
$$
which are different. Thus parentheses in nested <complex exponentiation> cannot be dropped or moved. There is an additional issue when all <complex logarithm> branches are allowed: each nested operation must retain its own branch choices. In the first expression, the complete set of values is $-ie^{-2\pi m}$, $m\in\mathbb Z$; in the second, it is $\exp[i(\pi/2+2\pi m)e^{-(\pi/2+2\pi k)}]$, $k,m\in\mathbb Z$. The principal-value comparison already establishes nonassociativity; the multivalued sets make the role of branching explicit.
Back to article page
