= Cotangent roots of a powered Möbius transformation
{title2=$z_k=-i\cot[(\pi/4+\pi k)/n]$}
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 $z=1$.
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