Question #188474

∮[(z - 1)/(z(z - t)(z - 3i)) dz] ; |z - i| = 1/2


1
Expert's answer
2021-05-12T01:52:12-0400

f(z)/(za)dz=2πif(a)\oint f(z)/(z-a) dz =2\pi i f(a)

[(z1)/z(zt)(z3i)]dz\oint [(z-1)/z(z-t) (z-3i)] dz

[(z1)/(z2tz)(z3i)]dz∮[(z−1)/(z^2 −tz)(z−3i)]dz

=[((z1)/(z2tz))/z3i]dz=2πf(a)=\oint [( (z-1)/(z^2-tz))/z-3i] dz = 2\pi f(a)=2πi[((t1)ln(zt))/(t23it)+((13i)ln(z3i))/(3it+9)+((i)ln(z))/3t]=2\pi i[ ((t-1)ln(|z-t|)) /(t^2-3it)+((1-3i)ln(|z-3i |)) /(3it+9) +((i) ln(|z|))/3t]




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