= Power-product map to a slit half-plane
{title2=$g(z)=(z-1)^k(z+1)^{1-k}$}
For $0<k<1$, principal upper-half-plane branches give a <conformal map> onto the upper half-plane cut along $[0,\tau]$, where $\tau=2k^k(1-k)^{1-k}e^{i\pi k}$. Its <derivative> is $(z-1+2k)(z-1)^{k-1}(z+1)^{-k}$. The two prevertices $-1,1$ map to the slit base, and $1-2k$ maps to its tip. Boundary traversal goes out along the slit and back, while $g(z)=z+1-2k+O(1/z)$ at infinity. The <argument principle> gives one preimage of every point off the slit.
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