CVCI1

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\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\,

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