Homotopy-invariance formulation of Cauchy's theorem

Statement

Suppose $U\subset \mathbb {C}$ 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 $f:U\to \mathbb {C}$, we have:

$\oint _{c_{1}}f(z)dz=\oint _{c_{2}}f(z)dz$

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

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.