From Complex analysis by Serge Lang.
Definition: $C_1,C_2:[a,b]\mapsto U$ are two closed paths in open set $U\subset \mathbb C$.Then we say $C_1$ is homotopic to $C_2$ if there exist a continuous function $h: [a,b]\times [c,d]\mapsto U$ such that
$$ h(t,c)=C_1(t) ~\text{and} ~h(t,d)=C_2(t)$$ for all $t\in [a,b]$. And $h(a,s)=h(b,s)$ for $s\in [c,d]$.
Now there is theorem in book
Theorem 5.2: If $f$ is holomorphic function in $U\subset \mathbb C$ open. And $C_1$ and $C_2$ are two closed homotopic path in $U$ then $$\int_{C_1}~f=\int_{C_2}~f$$.
Consider
$C_1:[0,2\pi]\mapsto \mathbb C$ By $\theta \mapsto e^{i\theta}$. $C_2:[0,2\pi]\mapsto \mathbb C$ By $\theta \mapsto 2e^{i(2\pi-\theta)}$ then
$$\int_{C_1}~\frac1z~dz=2\pi i \ne -2\pi i=\int_{C_2}~\frac 1 z~dz$$. Here $C_1$ and $C_2$ are homotopic in $\mathbb C \setminus {(0,0)}$. And $z\mapsto \frac 1 z$ is holomorphic in $\mathbb C \setminus {(0,0)}$. So we have all the hypothesis of theorem ,Then why both integral are not same ? Is homotopic path has same orientation ?