Homotopy-invariance formulation of Cauchy's theorem

Statement

Suppose $U \subset C$ is an open subset and $c_{1}, c_{2}$ are two cycles (cycle being a sum of smooth simple closed curves) that are smoothly homotopic. Then, for any holomorphic function $f : U \to C$ , we have:

$\oint_{c_{1}} f (z) d z = \oint_{c_{2}} f (z) d z$

In particular, if $c$ is a zero-homologous cycle, we have $\oint_{c} f (z) d z = 0$ .

Related facts

Goursat's integral lemma: It states something very similar, albeit in a very special case: where $c$ forms the smooth boundary of a region.