Complex integral depending on the chosen path?

Question

Let's say $f:\mathbb C\backslash\{0\}\to\mathbb C$ is holomorphic and $\text{Res}(f,0)=1$. Now if I look for example at the integral $$\oint_{|z|=2}f(z)dz$$ I get confused by the following: The path could for example be $\gamma(t)=2e^{it}$ where $t\in[0,2\pi]$, then the winding number of $\gamma$ around $z=0$ is exactly $1$ and the integral has the value $2\pi i$ according to the Residue Theorem. But can't we also use $\gamma(t)=2e^{-it}$ where $\gamma$ goes around $z=0$ clockwise and thus the winding number is $-1$? This would mean that the integral can also be $-2\pi i$. Where is my error?

Answer 1

There is no error on your part. It is just commonly agreed that this notation means you go around the circle exactly once in counter-clockwise direction. If you mean something else, you need to give the exact path.