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4. z = zo is a pole of order n for function f(z) if f(z) = (2_2) g(z) where g is analytic at zo and g(zo) * 0. Similarly, z = Zo is a zero of order n for function f(z) if f(z) = (z - Zo) g(z) where g is analytic at zo and g(Zo) # 0. 4.1. Show that if f(z) is analytic on a neighborhood of zo with a zero of order k > 1 at zo, then f'(z) has a zero at zo of order k - 1. (4 points) 4.2. Suppose f(z) is an analytic function close to zo. Let g(z) = f'(z) / f(z). Find the residue of g(z) at zo if f(z) has a zero of order m at zo. (4 points) 4.3. Suppose f(z) is an analytic function close to zo. Let g(z) = f'(z) /f(z). Find the residue of g(z) at zo if f(z) has a pole of order m at zo. (4 points)