Homotopy-invariance formulation of Cauchy's theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Suppose $U\subset \mathbb{C}$ is an open subset and ${\gamma }_1,{\gamma }_2$ are two smooth paths in $U$ such that there is a smooth homotopy between ${\gamma }_1$ and ${\gamma }_2$ preserving endpoints. Then, for any holomorphic function $f:U\to \mathbb{C}$, we have:

${\oint }_{{\gamma }_1}f(z)dz={\oint }_{{\gamma }_2}f(z)dz$

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.