Dirac $\delta$-function is widely used in science and engineering. Its Poisson kernel representation is of the form $$\eta_\epsilon(x)=\frac{1}{\pi}\mathrm{Im}\frac{1}{x-i\epsilon}=\frac{1}{\pi}\frac{\epsilon}{\epsilon^2+x^2},$$ which gives a $\epsilon^{-1}$ divergence as $x$ approaches zero. I was wondering about the case with a $\epsilon^{-1/2}$ divergence when $x$ approaches zero. For instance, consider the form $$\mathrm{Im}\frac{g(x)+i\epsilon-\sqrt{g(x)^2+2i\epsilon\:(i\epsilon+2)}}{f(x)+i\epsilon},$$ where real functions $f(x),g(x)$ vanish at $x=0$. When $x\rightarrow0$, the divergence is due to the square root part and leads to $\epsilon^{-1/2}$ divergence.
Does such $\epsilon^{-1/2}$ divergence have any relation to $\delta$-function? Is there any classification or description of such a kind?