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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Q: 12. h(x) =log, [log, (1+x)]

A: Please see the answer in Step 2

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Q: 12. If y=x, then y' is.. (HINT: Use logarithmic differentiation.)

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A: Here we use the property of composition of function.

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A: We have to check

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A: According to question,Given function,g(x) = ex - 1 - x + x2

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A: To find the derivative of  at   by  using Logarithmic differentiation.

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