MAA 4402 Final: complex-final-part1-fall2015
31 Jan 2019
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Maa 4402/5404 fall 2015 final exam part 1. Your work should be written in a proper and coherent fashion, and in a way that any student in the class can follow your work. [6 pts] solve the equation z3 = i, giving your (simpli ed) exact solutions in. Exhibit the solutions geometrically: [1 + 1 + 2 + 6 = 10 pts] (a) assume f (z) is a complex-valued function de ned on an open neighborhood of z0. What can you conclude about the di erentiability of f (z)? (d) suppose f (z) = z im z. Determine at which points f (z) di erentiable giving reasons. Find f (z) at these points: [4 + 4 + 4 = 12 pts] (a) de ne log(z). Sketch this domain. (b) show that log(( 1 + i)2) 6= 2 log( 1 + i). (c) assuming that log(z) is analytic on d, show that d dz. 1 z for z d: [6 pts]
