Labels:screenshot | circle | text | diagram | sketch | line | graphics OCR: Figure 12-17: Illustration of the Range of the Hyperbolic Tangent Function The diagram for tanh is simply that of tan turned on its ear: ślang = lanhiz. The imaginary axis is mapped onto itself with infinite multiplicity (period 2x), and the real axis is mapped onto the segment [-1, +]]: +00 is mapped to +1, and -go to -1. Vertical lines to the left or right of the real axis are mapped to circles surrounding -1 or 1, respectively. Horizontal lines are mapped to circular arca anchored at -1 and +1; two horizontal lines separated by a distance (2% + 1)x for integer & are together mapped into a complete circle. How do we know these really are circles? Well, tanh z = ((exp2z) - 1)/((exp 2z) +1), which is the composition of the bilinear transform (z - 1)/(z + 1), the exponential exp z, and the magnification 2z. Magnification mape lines to lines of the same alope; the exponential maps horizontal lines to circles and vertical lines to radial lines; and a bilinear transform mape generalized circles (including lines) to generalized circles. Q.E.D.
