Let $\Gamma$ be a curve in $\mathbb{C}$, parametrised by $\gamma:[a, b] \rightarrow \mathbb{C}$, that is $\gamma([a, b])=\Gamma$. Given $f: \Omega \subset \mathbb{C} \rightarrow \mathbb{C}$ and $\Gamma \subset \Omega$. > [!NOTE] Lemma > If $\tilde{\gamma}:[\tilde{a}, \tilde{b}] \rightarrow \mathbb{C}$ is another parametrisation of $\Gamma$ that preserves the orientation then $\int_{\tilde{\gamma}} f=\int_\gamma f.$ We refer to this fact as reparametrisation invariance. \[In practise, with the regularity we are demanding on the curves, this means that there exists $\phi:[\tilde{a}, \tilde{b}] \rightarrow[a, b]$, bijective and increasing, such that $\tilde{\gamma}=\gamma(\phi)$.\] ###### Proof We have $ \int_{\tilde{\gamma}} f=\int_{\tilde{a}}^{\bar{b}} f(\tilde{\gamma}(t)) \tilde{\gamma}^{\prime}(t) \mathrm{d} t-\int_{\tilde{a}}^{\bar{b}} f(\gamma(\phi(t))) \gamma^{\prime}(\phi(t)) \phi^{\prime}(t) \mathrm{d} t-\int_a^b f(\gamma(s)) \gamma^{\prime}(s) \mathrm{d} s=\int_\gamma f $ where we have made the change of variables $\phi(t)=s$ and therefore $\phi^{\prime}(t) \mathrm{d} t=\mathrm{d} s$.