Consider the fractional linear transformation in Exercise la above which maps 1 i to 0 2 to x and 0 to i 1 Without referring to

Question

Consider the fractional linear transformation in Exercise la above which maps 1 i to 0 2 to x and 0 to i 1 Without referring to an explicit formula determine the image of the circle 1 1 the image of the disk 1 1 and the image of the real axis Solution It is obvious that the tripple 1 1 2 0 lies on the circle 2 1 1 in the z plane From exercise II 7 1a we know that points in the w plane the tripple is mapped on i e 20 1 io 0 21 2 22 0 W W i 1 The circle 1 1 is mapped to the straight line through 0 and i 1 i e Im w Rew since tree points determines the circle By preservation of orientation the disk goes to the half plane to lower left of the straight line through 0 and i 1 Image of the real axis is circle orthogonal to image of 2 1 1 that must be the straight line through i 1 with slope 1 i e the line Im w Rew 2 II 7 2 z plane II 7 2 w plane

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