1. Working directly from the definition of a derivative, show that f(2)= 3zez + |2| +...

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1. Working directly from the definition of a derivative, show that f(2)= 3ze¹z + |2|² + 1 is differentiable at the origin and determine f'(0). 2. Assume a function f is analytic in an open set U. Let g(2) = f(). Show that g is analytic in the open set U* = {23 € U} and that g'(z) = f'(z)' for z € U*. 3. Show that f'(z) does not exist at any point z when a. f(2) = Re(z). b. f(2) = Im(z). 4. Find f" (2) when a. f(z) = iz +2 1. Working directly from the definition of a derivative, show that f(2)= 3ze¹z + |2|² + 1 is differentiable at the origin and determine f'(0). 2. Assume a function f is analytic in an open set U. Let g(2) = f(). Show that g is analytic in the open set U* = {23 € U} and that g'(z) = f'(z)' for z € U*. 3. Show that f'(z) does not exist at any point z when a. f(2) = Re(z). b. f(2) = Im(z). 4. Find f" (2) when a. f(z) = iz +2

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The question presented in the image explores the concepts of complex differentiability and the analy... View the full answer