solutionshw10-201141

solutionshw10-201141 - Homework 10 dz, using a branch of...

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Homework 10 (1) Evaluate R γ 1 z dz , using a branch of log z , where γ is the join of the line segments [1 - i, 1 + i ], [1 + i, - 1 + i ],and [ - 1 + i, - 1 - i ], starting at 1 - i and traversing the curve once (see figure 1). Figure 1. γ Solution: Let F ( z ) = log | z | + the branch of the log, such that θ arg z such that - π 2 < θ < 3 π 2 . Then R γ f ( z ) dz = F ( - 1 - i ) - F (1 - i ) = 3 πi 2 . (2) Compute Z 2 π 0 e cos t [cos(sin t + t )] dt and Z 2 π 0 e cos t [sin(sin t + t )] dt by computing R γ e z dz , where γ ( t ) = e it with 0 t 2 π . Solution: 0 = R γ e z dz = R 2 πi 0 e e it ie it dt = i R 2 πi 0 e e it + it dt = i R 2 πi 0 e cos t + i (sin t + t ) dt = i R 2 πi 0 e cos t [cos(sin t + t ) + i sin(sin t + t )] dt . Hence both integrals are 0. (3) Let α C with | α | 6
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