Finding potential function for $H(x,y)=(\frac{-y}{x^2+y^2}; \frac{x}{x^2+y^2})$ in $U=\mathbb{R}^2 \backslash (\{(x,0): x<0\} \cup \{(0,0)\})$

Question

My goal is to find one potential function $f:U \to \mathbb{R}$ for the vector field $H(x,y)=(\frac{-y}{x^2+y^2}; \frac{x}{x^2+y^2})$. I can define $f(x,y)=arctan(\frac{y}{x})$; $x\neq 0$ but now I should be able to extend my function $f$ to $U$ and thought of defining $f(x,y)=\frac{\pi}{2}; (x=0; y >0)$ and $ f(x,y)=-\frac{\pi}{2};(x=0: y<0)$. Any more suggestions?