Figure 4.15.7 illustrates the conformal mapping of the strip onto the whole -plane cut along the real axis from to and to , where and (principal value). Corresponding points share the same letters, with bars signifying complex conjugates. Lines parallel to the real axis in the -plane map onto ellipses in the -plane with foci at , and lines parallel to the imaginary axis in the -plane map onto rectangular hyperbolas confocal with the ellipses. In the labeling of corresponding points is a real parameter that can lie anywhere in the interval .
In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. See also About Color Map.
The corresponding surfaces for , , and are similar. In consequence of the identities
they can be obtained by translating the surfaces shown in Figures 4.15.8, 4.15.10, 4.15.12 by parallel to the -axis, and adjusting the phase coloring in the case of Figure 4.15.10.