By Lester R Ford
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Extra info for Automorphic Functions
The shaded regions are transformed into shaded regions as indicated by the arrows. For the elliptic transformation, (26) gives ~l + d)2 (a = 2 ~ e iO + e- iO = 2 + 2 cos o. (34) The second member is positive or zero and less than 4. a d is real and la dl < 2. From (27) we have + + a +d ± ( e1:-~2 + e -i~) 2 = = -+2 0 cos 2· Hence, (35) If 0 is commensurable with 7r, there will exist an integer n such that nO = 2m1r; and Kn = e2m1ri = 1. The result of applying the transformation n times is that each point is returned to its original position.
The point of tangency is then the fixed point. -We consider now the non-Ioxodromic transformations. Each" such transformation has a one-parameter family of fixed circles, including, as we found in Sec. 10, the line joining the centers of I and I'. The family of fixed circles is easily constructed. I t consists of the circles with centers on L orthogonal to I. itself by an inversion in I; and a reflection in L, a diameter, transforms it again into itself. Each fixed circle is also orthogo~al to I' from symmetry.
If P' coincides with PI two inversions are sufficient. Several alternative geometric transformations are possible. Thus, instead of inverting in I and then reflecting in L we may reflect in L and then invert in I'. Or we may rotate about - d/ c at the start; and so on. The preceding construction fails if I and I' coincide, for then L is not defined. In this case a = -d, or a d = 0; and T is an elliptic transformation of period two (Sec. 8). P' lies on I. An inversion in I followed by a reflection in L, the line joining -d/c to the midpoint of the arc PP' is equivalent to T.