= Solution
A sufficient <contour> setting for both results is a positively oriented, simple, closed, piecewise smooth <contour> $C$, with the function <holomorphic> on an open neighbourhood of $C$ and its entire interior. The <Cauchy integral theorem> then states
$$
\boxed{\oint_C f(z)\,dz=0}.
$$
If $z_0$ is strictly inside $C$, the <Cauchy integral formula> states
$$
\boxed{f(z_0)=\frac1{2\pi i}\oint_C\frac{f(z)}{z-z_0}\,dz}.
$$
The <contour> must not pass through $z_0$. Positive orientation means anticlockwise; reversing it reverses the sign of the <contour> integral. The holomorphy assumption throughout the interior excludes hidden singularities there.
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