 # 1. For each of the following functions, i) list its zeros (if any) ...

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1. For each of the following functions, i) list its zeros (if any) together with their orders, ii) list its singularities (if any) and for ench isolated singularity, state whether it is a removable singularity, a pole, in that case state its order or an essential singularity. (a) sin - :) (b) i 2. (a) Let C be the unit circle [=| = 1 positively oriented, and let / be given by /(z) = 1-22 How many times (indicate the direction, counterclockwise or clockwise) 2nd does the curve ((C) wind around the origin? Explain. (Hint: use the Argument principle). (b) Suppose / is analytic on [=| < 1, |/(x)| < 1 for all 2 in |=| < 1 and / has no zeros on [=| = 1. Find the number of solutions (counting multiplicity) of the equation /(z) = z" (n 1) in |=| < 1. Explain your answer. (Hint: Note that a solution of /(z) = z" is the same as a zero of z" - /(z). Then use Rouche's Theorern.) (2) 3. Suppose g is analytic and has a zero of order n at 20 (a) Show that the function of / given by (1(2)= has a pole of order n of 20. (b) What is the residue of at 2 = so? (Hint: take a look at the lecture notes on J'(2) /(x) Argument Principle).

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