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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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