Deformation of contours
Theorem 1.3 (Deformation of contours). Let \(C\) be a simple closed positively oriented contour, and let \(C_1, \dots , C_n\) be simple closed positively oriented contours inside \(C\) whose interiors are pairwise disjoint. If \(f\) is analytic on a domain containing all these contours and the region inside \(C\) but outside every \(C_k\), then
\[ \oint _C f(z) \dd z = \sum _{k=1}^{n} \oint _{C_k} f(z) \dd z. \]
Source: [1, ch. 4]