This form generates the de Rham cohomology group H^1_{dR}(\mathbf{R}^2\setminus\{0\}) \cong \mathbf{R}, meaning that any closed form \omega is the sum of an exact form df and a multiple of where k = \frac{1}{2\pi}\oint_{S^1} \omega accounts for a non-trivial contour integral around the origin, which is the only obstruction to a closed form on the punctured plane (locally the derivative of a potential function) being the derivative of a globally defined function.
