Math 417, Fall 2011Exercise Set #5Turn in: October 51.LetCdenote the line segment fromz=itoz= 1. Show thatCdzz4≤4√2.2.LetCRdenote the upper half circle for|z|=R, forR >1, parametrized in thecounter-clockwise direction. Show thatCRLog(z)z2dz≤2ππ+ lnRR.Conclude that the integral tends to zero asRtends to infinity.3.Suppose thatf(z) andg(z) are analytic functions defined in a domainΩ, and thatCis a contour inΩstarting atz1and ending atz2. Show that:Cf(z)g(z)dz=f(z2)g(z2)−f(z1)g(z1)−Cg(z)f(z)dzFor the next three problems, all integrals can be done using results from class or the
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