The following proof is the first hit for me on Google to find a proof that the fundamental group of the punctured plane is isomorphic to $\mathbb{Z}$: link
I've worked through it until the bottom of page 2 where it says:
"To prove this statement we consider the plane $C$ cut along $L$. The points of the cut have the parametrisation $R_+ \times [0, 2\pi]$ in polar coordinates."
Here $C$ is the complex numbers and $L$ is an arbitrary ray starting at $0$. My question is: how do they get $R_+ \times [0, 2\pi]$ for the points of the cut. If I cut along a ray I think the points should all be in $R_+ \times \phi_0$ for $\phi_0$ fixed.
Thanks for your help!