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Images of concentric circles around the origin Images of concentric circles of radii between  and  around the origin under the (conformal) map  . Circles near the origin are mapped near the points  . Images of the Riemann sphere of concentric circles of radii between  and  around the origin under the (conformal) map  . The images of the circles near to the south pole stay closed curves, while circles on the northern hemisphere are split into two parts because of the branch cuts along the real axis. Images in the  ‐plane of concentric circles of radii between  and  around the origin under the (conformal) map  . Images of radial rays originating from the origin Images of rays from the origin under the (conformal) map  . Images on the Riemann sphere of rays from the origin under the (conformal) map  . Images of a square grid Images of a square grid of size  under the (conformal) map  . Images of the tessellated Riemann sphere Images of triangles on the Riemann sphere under the (conformal) map  . Plane to plane map Curves  parametrized by  . Points  (where  ) move smoothly into points  .
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