Find a conformal map of the horizontal strip A Imz A onto the right half plane Rew 0 Hint Recall the discussion of the

Question

Find a conformal map of the horizontal strip A Imz A onto the right half plane Rew 0 Hint Recall the discussion of the exponential function or refer to the preceding problem Solution A Kumjian FIGURE II 6 4az FIGURE II 6 4aw It is clearly intended that A 0 so we will assume this We define the map h CC by h 2 e for all C Then h 2 2 for all C Hence h is analytic with a nowhere vanishing derivative and so h is conformal Note that the exponential functions maps the horizontal strip C 1 2 Im 2 onto the right half plane w C Rew 0 wC 2 Arg w 2 Hence h maps the horizontal strip C A Imz A onto the right half plane wC Rew 0

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