A small arc around a simple pole captures the same fraction of $2\pi i\Res$ as its angle captures of a full turn.Suggest a slogan
Let \(f\) be holomorphic on a punctured disc about \(z_0\in\CC\) with a simple pole at \(z_0\). For real numbers \(\alpha<\beta\) and \(\varepsilon>0\), let \(\gamma_\varepsilon\) be the arc \(t\mapsto z_0+\varepsilon e^{it}\), \(t\in[\alpha,\beta]\), of the circle \(\{z : \abs{z-z_0}=\varepsilon\}\). Then \[ \lim_{\varepsilon\to0}\int_{\gamma_\varepsilon}f(z)\dz=i(\beta-\alpha)\Res_{z=z_0}f. \]
