= Solution
Excise a small disc about zero and a thin slit along the negative real axis, using the orientations shown in the source: $C_1$ upwards, $C_3$ towards zero above the slit, $C_4$ clockwise around zero, and $C_5$ away from zero below the slit. On the resulting region $F$ is a single-valued <holomorphic function>; zero lies in the excluded disc and there are no <poles> in the region. <Cauchy's integral theorem> gives
$$
\boxed{\int_{C_1}F\,dz+\int_{C_2}F\,dz+\int_{C_3}F\,dz+\int_{C_4}F\,dz+\int_{C_5}F\,dz+\int_{C_6}F\,dz=0.}
$$
Here the paths on the cut mean limiting paths from its two sides. Apply <Cauchy's integral theorem> before taking those limits, when the entire <contour> lies inside the slit domain.
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