It is important in the last two theorems that the domain be simply connected. So this by showing that when U = {z :1< |z| < 3} that the function f(z) = z has no analytic logarithm. Hint: If e(2) z, then take a derivative to get eg(3) g'(2) = 1. This 1 implies g'(z) = e¬9(z) = ±. Let y be the circle |2| = 2. Then by the calculation we have just done 2ni. (you may assume that f, many times). Now explain why 2ni as we have done this calculation ) dz = 0 which gives a contradiction.

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