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.