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Problem Set #4 Due: 8 October 2024 at 23:59 1. Calculate all values of (i) (V/3 -2)7; (ii) (-1 + 2) V3 2. Let tan z = sing be the function "tangent" of a complex argument z. It is not injec- tive, so its "inverse" arctan z is defined as the multivalued function that satisfies the equation tan(arctan(z) ) = z. (i) Express arctan z in terms of log z. (ii) If g(z) is a branch of arctan z which is continuous at zo, prove that g(z) is differentiable at zo and find g'(zo). (iii) Show that Arctan z := > Log it is a brach of arctan z and find the region D C C in which Arctan z is defined. 3. Find a branch of the multivalued function f(z) = (23 + 1) 3 defined (and holomorphic) in the region D = C\\(I1 U I2 U 13), where I1 = [-1, 0] is the interval on the real line, 2 is the closed interval connecting the origin with the point ? + i?, and 13 is the closed interval connecting the origin with the point ? - 1 2. Express your answer in terms of the principal brach of the log-function. Hint. Study the map 4 : Cx - C defined by p(z) = 1 + 3 is a 3-to-1 map from D onto C\\Rso