 Cotangent roots of a powered Möbius transformation

For a nonzero integer $n$, the equation $((z+1)/(z-1){)}^n=i$ has $|n|$ distinct roots given by the displayed formula for $0\le k<|n|$. Put $w=(z+1)/(z-1)$; its roots have unit modulus and angles $(\pi /2+2\pi k)/n$. The inverse map is $(w+1)/(w-1)=-i\cot (\arg (w)/2)$, proving that all these roots are purely imaginary. The value $w=1$ never occurs. For $n=0$ there are no solutions; for negative $n$ the original map's zero at $z=-1$ is excluded as well as its pole at $z=1$.