Answer to Question #188474 in Calculus for John

Question

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

Answer 1

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

$\oint [(z-1)/z(z-t) (z-3i)] dz$

$∮[(z−1)/(z^2 −tz)(z−3i)]dz$

$=\oint [( (z-1)/(z^2-tz))/z-3i] dz = 2\pi f(a)$$=2\pi i[ ((t-1)ln(|z-t|)) /(t^2-3it)+((1-3i)ln(|z-3i |)) /(3it+9) +((i) ln(|z|))/3t]$