HOMEWORK 3 DUE MARCH 6 1. Let f : C1 ! C1 be the Mobius transformation f (z) = 1z . Find the map g : S2 ! S2 on the unit sphere that corresponds to f via the stereographic projection. 2. Map the region between the circles jzj = 1 and z ?? 12= 12 to the upper half plane fz : Im (z) > 0g. 3. Map the region G = fz : Im (z) > 0 and jzj > 1g to the upper half plane. 4. Find Rp1z dz where (a) is the upper half of the unit cicle from +1 to ??1 (b) is the lower half of the unit cirlce from +1 to ??1. 5. Prove that lim r!1Zr eiz z dz = 0 where r : [0; ] ! C is r (t) = reit: 6.
HOMEWORK 3 DUE MARCH 6 1. Let f : C1 ! C1 be the Mobius transformation f (z) = 1z . Find the map g : S2 ! S2 on the unit sphere that corresponds to f via the stereographic projection. 2. Map the region between the circles jzj = 1 and z ?? 12= 12 to the upper half plane fz : Im (z) > 0g. 3. Map the region G = fz : Im (z) > 0 and jzj > 1g to the upper half plane. 4. Find Rp1z dz where (a) is the upper half of the unit cicle from +1 to ??1 (b) is the lower half of the unit cirlce from +1 to ??1. 5. Prove that lim r!1Zr eiz z dz = 0 where r : [0; ] ! C is r (t) = reit: 6. Let f be analytic in a region and suppose that jf (z) ?? 1j 0 and an integer n 1 so that jfj (z) M jzjn ; for all jzj 1. Prove that f is a polynomial of degree n. 8. Let f be an entire function so that jfj (z) 1 for all z 2 C with jzj 1. Show that f is a polynomial. 9. Let f be an analytic function dened in a neighborhood of the origin in C so that f (0) = 0 and f0 (0) 6= 0. Fix an integer n 1. Show that there exists an analytic function dened in a neighborhood of the origin so that f (zn) = g (z)n ; for all z. 10. Prove that any positive harmonic function u : R2 ! R must be a constant. 1
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BENCHMARK – EFFECTS OF CHILDHOOD TRAUMA WORKSHEET
Academic Level University Subject Healthcare Type of Paper Other (Not listed) Paper Format APA Assessment Traits Benchmark Requires Lopeswrite Assessment Description Complete the “Effects of Childhood Trauma Worksheet” document attached. While APA format is not Read more…