Question

Due by 12:00 noon, today 05/12/20. : CU{co} → Problem 1. Consider the Möbius transformation CU{o} defined by S(z) = 171 (i) Compute f(1), f(), f(-1), f(-i). (ii) Show that for 2 = ei, where 0 ER, f(x) is real or oo, that is f(el) E RU{0} (iii) Let D = {z zz < 1} denote the open unit disc and H = {z | Im(2) >0} denote the upper half plane. Show that f takes D onto H. (iv) Find the power series expansion of f centered at zero and com- pute its radius of convergence. Problem 2. Let rı = e271/3 and r2 = F = e-2/3 (i) Verify that i andry are the two roots of the quadratic equation wa+w+1=0 (ii) Verify the following decomposition into partial fractions: -14 ) w2 + w +1 ri-r2 'w-ri W-T2 (iii) Let y be the loop defined for 6 € (0,21] by 7(0) = 2et. Compute the path integral 1 Jy w²+w+1 ar

Answer 1

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