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(Solved): (10 points) Consider the following paths: \gamma _(1)(t)=2e^(it),0<=t<=(\pi )/(2), \gamma _(2 ...
(10 points) Consider the following paths:
\gamma _(1)(t)=2e^(it),0<=t<=(\pi )/(2),
\gamma _(2)(t)=2i-(1+2i)(t-(\pi )/(2))*(2)/(\pi ),(\pi )/(2)<=t<=\pi ,
\gamma _(3)(t)=2e^(it),\pi <=t<=(3\pi )/(2),
\gamma _(1)(t)=-2i+(1+2i)(t-(3\pi )/(2))*(2)/(\pi ),(3\pi )/(2)<=t<=2\pi Compute (a)
\int_(|z|=R) (1)/(z)dzfor all
R>0(b)
\int_(\gamma _(1)) (1)/(z)dz+\int_(\gamma _(2)) (1)/(z)dz+\int_(\gamma _(3)) (1)/(z)dz+\int_(\gamma _(4)) (1)/(z)dz