Solution
= Solution
Off the slit choose the stated analytic square-root branch. <Differentiation> gives $f\prime(z)=c^{-1}(1+z/\sqrt{z^2-c^2})$, which does not vanish there, so the map is conformal.
= Solution
Off the slit choose the stated analytic square-root branch. <Differentiation> gives $f\prime(z)=c^{-1}(1+z/\sqrt{z^2-c^2})$, which does not vanish there, so the map is conformal.