hw11-201141 - at the points of intersection of the boundary...

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Homework 11 (1) Evaluate (without parametrizing, but using Cauchy’s Integral Theorem) R γ 1 1+ z 2 dz for a. γ ( t ) = 1 + e it (0 t 2 π ). b. γ ( t ) = - i + e it (0 t 2 π ). c. γ ( t ) = 2 e it (0 t 2 π ). d. γ ( t ) = 3 i + 3 e it (0 t 2 π ). (2) Let α C with | α | 6 = 1. Compute Z 2 π 0 dt 1 - 2 α cos t + α 2 by computing i α Z γ 1 ( z - α )( z - 1 α ) dz, where γ ( t ) = e it with 0 t 2 π . (3) Compute Z 2 π 0 cos t 5 - 4 cos t dt. (4) Which of the following sets are starlike? In case they are starlike, find a star center of the set. a. { z C : | z | < 1 and | z + 1 | > 2 } , b. { z C : | z | < 1 and | z - 2 | > 5 } , (Hint: Both sets are “sickle shaped”. To find possible star centers, find the point of intersection of the tangent lines to the boundary curve closest to the origin
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Unformatted text preview: at the points of intersection of the boundary curves. ) (5) (Quals 1998) Let G C be an open set containing the closed disk B ( a ; r ) = { z : | z-a | r } . Let h f n i be a sequence of holomorphic functions on G such that f n ( z ) 0 uniformly on { z : | z-a | = r } . Prove that f n ( z ) 0 for all z in the open disk B ( a ; r ). (Hint: Use Cauchys Integral Formula.) 1...
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