CVCI1

$\oint_{|z|=2} \frac{1-2z}{z(z-1)(z-3)} dz\,$

$=\oint_{|z|=2} \frac{1-2z}{z^3-4z^2+3} dz = 2\pi i \sum \mathrm{Res}\,$

The integrand has simple poles at $z=0,1,3\,$ . Only 0 and 1 are within the contour, so

$\oint_{|z|=2} \frac{1-2z}{z(z-1)(z-3)} dz = 2\pi i (\mathrm{Res}_{z=0} + \mathrm{Res}_{z=1} )\,$

$=2\pi i\left(\frac{1-2z}{3z^2-8z+3}\Big|_{z=0} + \frac{1-2z}{3z^2-8z+3}\Big|_{z=1}\right)\,$

$=2\pi i \left(\frac{1}{3} + \frac{1}{2}\right) = \frac{5}{3}\pi i\,$